Factorizations, injectivity and compactness in categories of modules

This article was downloaded by: [University of Illinois Chicago]On: 16 April 2013, At: 09:08Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/lagb20

Factorizations, injectivity andcompactness in categories ofmodulesDikran Dikranjan a & Eraldo Giuli ba Institute of Mathematics, Bulgarian Academy of Sciences,Sofia, 1090, Bulgariab Dipartimento di Matematica Pura ed Applicata, UniversitáStudi di L'Aquila, L'Aquila, 67100, ItaliaVersion of record first published: 27 Jun 2007.

To cite this article: Dikran Dikranjan & Eraldo Giuli (1991): Factorizations, injectivity andcompactness in categories of modules, Communications in Algebra, 19:1, 45-83

To link to this article: http://dx.doi.org/10.1080/00927879108824129

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make anyrepresentation that the contents will be complete or accurate or up to date. Theaccuracy of any instructions, formulae, and drug doses should be independentlyverified with primary sources. The publisher shall not be liable for any loss, actions,claims, proceedings, demand, or costs or damages whatsoever or howsoever causedarising directly or indirectly in connection with or arising out of the use of thismaterial.

http://www.tandfonline.com/loi/lagb20

http://dx.doi.org/10.1080/00927879108824129

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

COMMUNICATIONS I N ALGEBRA, 1 9 ( 1 ) , 45-83 ( 1 9 9 1 )

FACTORIZATIONS, INJECTIVITY AND COMPACTNESS IN CATEGORIES OF MODULES

Dikran ~ikranjan*

Institute of Mathematics Bulgarian Academy of Sciences

1090-Sofia, Bulga~ia

Eraldo ~ i u l i * *

Dipartimento di Matematica Pura ed Applicata Universiti degli S t u d di L'Aquila

67 100-L'Aquila, Italia

ABSTRACT : A notion of closure operator for modules is used to characterize factorization structures in categories of modules. Moreover compactness, injectivity and absolute closedness are studied with respect to such closure operators. A criterion for compactness of modules is obtained in terms of injectivity or absolute closedness of the quotients extending recent results of Temple Fay.

AMS Subj. Class.: 16A22, 16A52, 16A90, l8A20, 18A32, 16A63, l8E40.

preradical, torsion theory, closure operator, factorization structure, C-in:jective module, absolutely C-closed module, C-compact module.

* The first author acknowledges support from Italian National Research Council. * * The second author acknowledges a grant from the Italian Ministry of Public Education.

Copyright O 1991 by Marcel Dekker, Inc.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

DIKRANJAN AND G I U L I

Introduction

Let r be a torsion theory in the category R-Mod of left modules over a fixed ring

R with unity. Then every homomorphism f has a standard factorization into a

composition f = me of a morphism e with r-torsion cokernel and a monomorphism

m with r-torsion-free cokernel which is essentially unique. On the other hand this

torsion theory defines a "closure operation" : a submodule N of a module M is r-

closed in M if M/N is r-torsion-free and N is r-dense in M if M/N is r-torsion. In

this terms the morphisms e and m in the above factorization are r-dense and r -

closed respectively. The torsion theory r determines also a relative concept of

injectivity - r-injective modules (with respect to r-dense monomorphisms) and a

weaker version - the absolutely r-closed (absolutely r-pure) modules, i.e r-torsion-

free modules which are r-closed in each r-torsion-free module Observe that the

factorization structure, r-injectivity and absolute r-closedness depend only on the

closure operation defined by r. Therefore it is natural to build the above theories for

an abstract closure operator. This is one of the aims of the present paper. We make

use of the abstract theory of closure operators developed in a quite general situation

in [DiGi 871, where factorization structures were shown to be essentially closure

operators with very special features.

The main external characterization of compact topological spaces is given by the

Kuratowski-Mrowka theorem: a space X is compact if and only if the projection p : XxY + Y is a closed map (i.e. sends closed sets to closed sets) for each space Y.

Compact objects with respect to a factorization structure were introduced in [Ma 741 and [HeSaSt 87) in a general setting. In the case of topological spaces and

the usual factorization of a continuous maps into the composition of a map with

dense image and a closed embedding, this gives the Kuratowski-Mrowka

Theorem. Fay [Fa 881 characterized the compact modules with respect to the

standard factorization structure corresponding to a hereditary torsion theory r in

terms of r-injectivity. This result was estended in [FaWa 891 to the case of

nilpotent groups and a particular r.

The main purpose of this paper is to introduce and study the notion of compact

module with respect to an arbitrary closure operator C of R-Mod - C-compact

module. Such an approach to compactness in topological categories was adopted in

[DiGi 88bI. In order to make the paper accessible to readers in both domains, in

Section 1 we recall some definitions and standard properties of preradicals and

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZATIONS, INJECTIVITY AND COMPACTNESS 47

torsion theories (cf. [La 711, [BiKeNe 821, [Go 861) and in Section 2 we give

a self-contained exposition of the theory of closure operators of R-Mod (cf. [DiGi

871). To each closure operator there corresponds a preradical while to each

preradical there corresponds a complete lattice of closure operators with the same

preradical. We distiguish two of them - the greatest and the smallest element of this

lattice -ant1 call them respectively the maximal and the minimal closure 3perator

corresponding to the preradical. The first one, in the case of an idempotent radical

r , is the closure operator described above. We consider also closure operators

generated by couples of preradicals.

In Section 3 we discuss factorization structures of R-Mod and their -elations

with closure operators and preradicals (Proposition 3.1 and Corollary 1.3). We

give many examples of non-standard factorization structures and characteirze the

preradicals admitting only the standard factorization structure (Corollary 3.4).

In Section 4 C-injectivity and absolute C-closedness relying on dertsity and

closedness with respect to a closure operator C are introduces and studied. We give

examples to ditinguish these notions (Example 4.4, Remark 4.8), which coincide

in the case of weakly hereditary closure operators C with hereditary preradical

(Proposition 4.7). In the case of a maximal closure operator the latter r e s ~ l t is well

known ([GO 861). Absolutely C-closed objects were studied already by [sbell [Is 661 in the category of semigroups and by the authors in categories of to.~ological

spaces ([DiGi 88a1, [DiGi 88b], [DiGiTh 901).

Section 5 is devoted to C-compactness. We show that for arbitrary prersdical r a

module is r-compact if and only if all r-torsion-free quotients are absolutely r -

closed (Corollary 5.1 1). We show also that this is the right extension of Fay's

characterization in the general case. In the case of the typical non-hereditary torsion

theory r in the category of abelian groups determined by the maximal divisible

subgroup we show that every cotorsion abelian group, in particu ar every

algebraically compact group, is r-compact. Moreover r-compact groups which are

Hausdorff in their natural topology are cotorsion. We pay some attention also to a

relative notion of C-compactness and productivity for C-compact modules.

The major part of what we have done for modules can be done in abelian

categories. The technique based on closure operators developed in Sections 2 and 3

will be used in a forthcoming paper dealing with epimorphisms and corrpleteness

in categories of modules [DiGi 901

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

48 D I K R A N J A N AND G I U L I

It is a pleasure to thank Adalberto Orsatti for his helpful suggestions regarding

Theorem 5.22 and Corollary 5.23. The first (larger) version of this paper circulated

among friends and colleagues. These results were exposed also in talks given by

the f i s t named author at York University, and by the second named author at the

University of Cape Town. Our thanks are extended also to these universities and

colleagues.

1. Preliminaries

We denote by N the set of naturals, by 2. - the integers, by Q - the rationals The

categorical terminology is that of [HeSt 791.

A preradical r in R-Mod is a subfunctor of the identity functor of R-Mod. It is

called

idernpotent if r(r(M)) = r(M) for each module M;

radical if r(M/r(M)) = 0 for each module M; hereditary if r(N) = r(M)nN for each M E R-Mod and for each submodule N

of M; cohereditary if r(M/N) = (r(M)+N)/N for each M E R-Mod and for each

submodule N of M.

It is easy to verify that every hereditary preradical is idempotent and that every

cohereditary preradical is a radical. A preradical r is cohereditary if and only if there

exists an ideal I of R such that r(M) = IM for each M 6 R-Mod; clearly then I =

r(R). In such a case r is idempotent if and only if I = I ~ . If R = Z and n is a

natural number, we denote by n the cohereditary radical determined by the principal

ideal generated by n. A partial order in the conglomerate PR of all preradicals in R-Mod is defined by

r I s if r(M) c s(M) for each M E R-Mod. PR is a complete (illegitimate) lattice

with respect to the sum C and intersection n of preradicals defined by (Cri)(M) =

Cri(M) and (nri)(M)= n r i (M) respectively. Sums and intersections are well

defined for arbitrary classes of preradicals since in R-Mod the monomorphisms

into a fixed module are essentially a set ,i. e. R-Mod is well-powered.

For preradicals r and s the composition is denoted by r s and (r:s) denotes the

preradical defined by the conditions: (1) (r:s) 2 s , (2) (r:s)(M)/s(M) = r(M/s(M))

for each module M.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZATIONS, INJECTIVITY AND COMPACTNESS

Every preradical r defines two series of preradicals by setting

r1 = r , roc+'= (r:roc) and rP =C(rOL: a < (3)

r = r = r r a , rp = n ( r , : o: < p), for every ordinal o: and every limit ordinal (3. Then r* = CrOL is the smallest

radical containing r and r, = nr, is the largest idempotent radical contained in r .

They will be called radical hull and idempotent core , respectively, of the xeradical r . The preradical r, is a radical if r is a radical, while r* is idempotent (hcreditary)

if r is idempotent (hereditary).

To each preradical r a torsion-flee class F, = {M : r(M) = 0) and a torsion class

3, = (M : r(M) = M ) are associated. The following properties are well known.

PROPOSITION 1.1. Let r be a preradical. Then: (a) for every family (Mi : i E I ) of modules the following hold

r (n{Mi : i E I ) ) c n {r(Mi): i E I ) and r(QMi) = Br(Mi'1.

(b) F, is closed under the formation of products and submodules;

(c) 3, is closed under the formation of sums and quotients;

(d) If r is a radical and N is a submodule of M E R-Mod containel in r(M)

then the canonical morphism q : M -. M/N satisfies q(r(M)) = r(M/N);

(e) If r is idempotent (resp. a radical) then F, (resp. 7,) is closed also with

respect to extensions.

If r is an idempotent radical, then the pair (Tr, F r ) is called torsion theory

generated by r . Conversely, each pair (3, F) of classes of modules, such that 3 is

closed with respect to sums, quotients and extensions, F is closed with respect to

products, submodules and extensions, determines an idempotent radical r. such that

7 = T r a n d F = F r .

We note that, for an arbitrary preradical r , r(M) = 0 implies r*(M) = 0, and r(M) = M yields r,(M) = M. Consequently Fr* = Fr, and Tr, = Tr, while in general

Fr c Fr, and Tr c Tr*.

In general we obtain two torsion theories from a preradical r. he) correspond to the idempotent radicals (r,)* and (r*), which satisfy always (r,)* 5 (re), and do

not coincide in general (cf. Example 1.3.(d) below). They coincide whmever r is

either idempotent or a radical.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

50 DIKRANJAN AND GIULI

Every class A of modules generates a radical rA by setting for a module M,

If 4 is closed under the formation of products and submodules then M/rA(M)

belongs to A, i.e. there exists an epimorphism f : M + N such that N belongs to A and Ker f = rA(M). In view of the future application we give separately the

following result.

PROPOSITION 1.2. The assignements r + Fr, A + r, determines a Galois

correspondence between PR and the conglomerate of all subclasses of R-Mod,

which is a Galois equivalence between radicals and classes of modules closed

under the formation of products and submodules.

The above Galois correspondence also determines a Galois equivalence between

idempotent radicals and classes of modules closed under the formation of products,

submodules and extensions.

EXAMPLES 1.3. (a) Let z(M) denote the maximal singular submodule of a module

M. Then z is a hereditary preradical; the torsion theory associated to z* = z2 is

usually known as Goldie torsion theory.

If R = Z then z(M) is the subgroup of all torsion elements of the abelian group M.

It is usually denoted by t(M).

(b) Let P be a family of painvise non-isomorphic simple modules. For a module M denote by Soct(M) the sum of all simple submodules of M isomorphic to some

of the modules in P. This defines a hereditary preradical. If P is a full family of representatives of the simple modules then S o c ~ is denoted simply by Soc. The

torsion theory generated by Soc is known as Dickson torsion theory. For R = Z and a set IP of prime numbers denote by sp the preradical SOCK, where

P = (Z/pZ : p E IP). Its radical hull is the sum t p of all p-torsion components of

M for p E IP. It is a hereditary radical which fails to be cohereditary. This is the

typical form of a hereditary radical in the category of abelian groups.

(c) Let R be a left hereditary ring (then quotients of injective modules are

injective). If R is also left noetherian then the class 1 of all injective modules is the

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


torsion class of a torsion theory. We denote by d the radical corresponding to this

torsion theory. If R = Z then d(G) is the maximal divisible subgroup of the

abelian group G.

For a prime number p denote by dp(G) the maximal p-divisible subgroup of G

and if IP is a set of prime numbers denote by d p the sum of all radicds dp for

p~ IP. Then both dp and d p are idempotent radicals which are neither hrzreditary

nor cohereditary.

(d) For IP ZI IP' nonempty sets of prime numbers consider r = Socpdp'(see

(b) and (c)). Then r, = 0 hence (r,)* = 0. Indeed rx = 0, since IP 3 I?' yields

d p 4 o c p = 0. On the other hand (r*), * 0. In fact, note first that f p d p r =

t p n d p f since t p is hereditary, so that t p d p is a radical and it coincides with r*.

Moreover r* is idempotent since dp ' tp = tpdpt; finally dp ' tp * 0 since IP and

IP' are nonempty. Note that for each n, n, 2 d. Moreover dp = p, and more generally, if' IP, is the

set of all primes dividing n then d p = n,.

(e) For each module M denote by a(M) the intersections of all maxirr:ol proper

submodules of M. Then a is a radical which is not idempotent (wkence not

hereditary).

2. Closure operators and preradicals

Recall from [DiGi 871 that a closure operator of R-Mod (with respect to

monomorphisms) assignes to each submodule N of an arbitrary module M a submodule CM(N) of M such that, for each pair of submodules N, P of M and for

each homomorphism f : M + M', the following conditions are satisfied:

(c,) N c C h ? ( N )

(c,) N c P => CM(N) c CM(P)

(cg) f(CM(N)) c CM!(f(N)) (continuity property).

A closure operator C is called:

weakly hereditary if CCM(N)(N) = CM(N) for each N c M;

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

5 2 DIKRANJAN AND GIULI

hereditary if CN(L) = CM(L)nN for each L c N c M;

idernpotent if CM(CM(N)) = CM(N) for each N c M.

As a simplest example one should consider the discrete closure operator defined by CM(N)= N for each M and N. Clearly it has all three properties mentioned

above.

Pierce [Pi 701 introduced and studied "closure operators" in R-Mod in a different sense: (i) the closure is defined for subsets, (ii) (c3) is not required, (iii)

the closure is additive for countable chains.

Let C be a closure operator of R-Mod. A submodule N of M is called C-closed (C-dense) in M if CM(N) = N (respectively, CM(N) = M)), and a homomorphism

f:N + M is called C-dense if CM(f(N)) = M.

The conglomerate 60 of all closure operators of R - M o d is a complete

(illegitimate) lattice with respect to the order defined by C I D if and only if CM(N)

I DM(N) for each M E R-Mod and submodule N of M. The join v and the meet A

in @O are given by (vCi)M(N) = C(Ci)M(N) and (r\Ci)M(N) = n(Ci),(N), the

discrete closure operator is the smallest element of 60. If (Ci) is a family of

weakly hereditary (idempotent) closure operators, then vCi (AC~) is also weakly

hereditary (idempotent) (cf. [DiGi 871).

For closure operators C and D we denote by CD their composition defined by (CD)M(N) = CM(DM(N)) for every module M and submodule N of M. We define

also another (internal) composition C#D by setting

A straightforward verification shows that both compositions are closure operators.

For every closure operator C one defines an ascending chain of closure operators

Ca by

c '=c, ca+'=cca and c P = v ( c a : o : < ( 3 )

for every ordinal o: and every limit ordinal (3. So C* = v ( C a ) is an idempotent

closure operator and for every idempotent closure operator D 2 C one has D 2 C*.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZ4TIONS, INJECTIVITY AND COXPACTNESS 5 3

C* will be called idempotent hull of C. It is weakly hereditary provided C is

weakly hereditary (cf. [DiGi 87)). One also has a descending chain C, of closure operators defined by

C 1 = C , C,+,=C#C, and C p = ~ { C Q : : a < P I .

So C* = A{C,] is a weakly hereditary closure operator and for evei-y weakly

hereditary closure operator D I C one has D I C*. The closure operator Cl* will be

called weakly hereditary core of C. It is idempotent whenever C is (cf. [DiGi 871). This fact implies that (C*), is idempotent, consequently the ineq~.ality C*

I (C*)* yields (C*)* I (C*)*. We show below that these closure operators need

not coincide in general.

Let C be a closure operator of R-Mod. For each module M set

By (c3) it follows that rC is a preradical of R-Mod. The corres~ondence

cD : CO +. PR defined by @(C) = rC has the following properties:

PROPOSITION 2.1. (a) For each (eventually large) class (Ci] in CO the -‘allowing

(b) if C is (weakly hereditary) hereditary, then @(C) is (idempotent) hereditary.

(c) for closure operators C and D we have

Q(C#D) = @(C)@@) and Q(C), = @(C,) for each ordinal (2.

In particular @(C), = @(C,).

PROOF. (a) and (b) follow directly from the definitions.

(c). The first part follows directly from the definitions, the second piLq follows

from (a) and the first part by transfinite induction.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


Clearly C is the discrete closure operator if and only if @(C) = 0. Thus (c) above

yields that C* is discrete if and only if cf,(C)*=O.

By (a) cf, is monotone, in other words it is a functor if we consider the partially

ordered classes OD and PJR as categories in the usual way.

Now we see that the idempotency of C gives no information on cf,(C). Let r be a

preradical of R-Mod. Define, for each module M and each submodule N of M,

It is easy to see that C, is an idempotent closure operator of R-Mod and that

(D(C,) = r. If C is a closure operator of R-Mod then, by (c2), we have Co(C) I C.

A closure operator C of R-Mod will be called minimal if it satisfies the condition C=Co(C) . Then the assignement r + C, is an order preserving bijection between

the conglomerate PR of all preradicals of R-Mod and the conglomerate Mhm of

all minimal closure operators of R-Mod, whose inverse is the restriction of cf, on MinCO.

The above observation and Proposition 2.1 yield

PROPOSITION 2.2. MinnCO is a complete v-subsemilattice of CO. The

assignement r -+ C, is a join-semilattice isomorphism between PX and MinCO.

PROOF. Let ( r i : i s I ) be a family of preradicals and let r = C ( r i : i s I ) . For each

M E R-Mod and N c M we have

(C,)M(N) = N+C{ri(M) : i~ I ) = C{N + ri(M) : k I ) =

X{ (CriIM(N) : i~ 11 = (v ( K r i ) : i~ U)M(N).

The following observation helps us to define another closure operator associated

to a given preradical. Let C be a closure operator of R-Mod. Then for every

module M and every submodule N of M consider the canonical homomorphism

q : M + M/N. By the continuity of C we have

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZ4TIONS, INJECTIVITY, AND COMPACTNESS 55

The closure operator C will be called maximal if in (2) equality holds for each

module M and each submodule N of M. In such a case C is uniquely determined by

@(C) and (2), since then CM(N) = q- l (cD(c)(~/N)) (such operators were called

standard in [DiGi 871). Denote by Cr the maximal closure operator satisfying

cD(Cr) = r , i.e. for every preradical r set

Then the assignement r + Cr is an order preserving bijection beiveen the conglomerate PR of all preradicals of R-Mod and the conglomerate M80tC0 of all

maximal closure operators of R-Mod, whose inverse is the restriction of cD on

MaxCO.

PROPOSITION 2.3. hkitcQ is a complete sublattice of %o and the assignement

r + Cr is a complete lattice isomorphism between PR and Mum.

PROOF. Let (r i : iEI] be a family of preradicals and let r = x ( r i : i~ I ] . For each

M E R-Mod and N c M, by definition r(M/N) = C(ri(M/N) : iEI], so we have

(Cr), (N) = q - l ( r ( ~ / ~ ) ) =q-l(x{ri(M/N) : i s ] ) ) =

z{q-l(ri(M/N)) : i~ I ] = (v(Cri:i€ 1]IM(N).

Analogously for intersections.

As a particular case of Proposition 2.2 and 2.3, C, (resp. Cr ) is discrete if and

only if r := 0. Notice that the assignements r + C, and r -t C' are respectively right and left

adjoint of the functor Q.

PROPOSITION 2.4. (a) The assignement r -t Cr agrees with both corn ~ositions,

more precisely CrS = Cr#CS and c (~ : s ) = C T S ;

(b) For each maximal closure operator C and any D we have cD(CD) = (cD(C):cD@)) and cD(C)OL=Q(COL).

In particular cD(C)* = cD(C*). (c) ([DiGi 87, Ex. 6.11) For each preradical r and ordinal we haw

(C'), = and (cr )Q = ~ ( r " ) .

In particular (Cr)* = Cr* and (Cr)* = cr*.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


PROOF. (a). It suffices to prove that, for every module M and submodule N of M,

the equalities (Cr#CS)M(N)/N= (CrS)M(N)/N and

(CrCS)M(N)/(CS)M(N) = ( C ( ~ : ~ ) ) ~ ( N ) / ( C ~ ) , ( N ) hold. In fact (Cr#CS)M(N)/N = (C')(C~)M(N)(N)/N = r((CS)M(N)/N) =

r(s(M/N)) = (rs)(M/N) = (CrS)M(N)/N. This proves the first equality. For the

(b). The first part follows directly from the definitions and the second part

follows from Proposition 2.l.(a) and the first part by transfinite induction.

(c). It follows from (a) and Proposition 2.3 by transfinite induction that, for every ordinal a and every maximal closure operator C, both COC and C, are

maximal closure operators. Now i n virtue of Proposition 2.l.(c), for C = Cr, Q(C,) = @(C), and by (b), @(Ca) = @(C)&, i.e. (b) holds.

If r is a preradical we abbreviate Cr-closed to r-closed and Cr-dense to r-dense.

We note that for r-closed the term r-pure is also used when r is an idempotent

radical (cf. [Go 861). By (c) of the above proposition r-closedness and r* - closedness coincide, while r-density coincides with r*-density.

It was observed before that (C*)* 2 (C*)* holds for each closure operator C.

This inequality may be proper. In fact, it follows from Proposition 2.4.(b) that, for

r as in Example 1.3.(d), the maximal closure operator C corresponding to r satisfies (C*)* < (C*)*.

In the following proposition we give some relations between the properties of the

maximal and minimal closure operators and the corresponding preradicals.

PROPOSITION 2.5. (a) A maximal closure operator C is idempotent if and only if

@(C) is a radical.

(b) Assume C either minimal or maximal; then C is (weakly hereditary)

hereditary if and only if @(C) is (idempotent) hereditary.

(c) The idempotent hull of every maximal and hereditary closure operator C is

hereditary as well.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

F A C T O R I Z A T I O N S . I N J E C T I V I T Y AND COMPACTNESS 5 7

PROOF. (a'). Follows from Proposition 2.4.(c).

(b). The necessity follows in both cases frorn Proposition 2.l.(b). Assurne now that r = @(C) is an idempotent preradical. Then for each M E R-Mod a ~ d each

submodule N of M we have (C~) (C , )~ (N) = N+r((C,)M(N)) = N+r(N+r(M)) 3

N+r(r(M)) = N+r(M) = ( C r ) ~ ( N ) . This proves the sufficiency in case C = C, is a

minimal closure operator. In case C is maximal, Proposition 2.4.(c) ap3lies. If @(C) is her(-ditary, then the hereditarity of C is checked directly.

(c). By F'roposition 2.3, C = Cr for a (unique) preradical r and (cr)* := c r * by

Proposition 2.4.(c). On the other hand, by (b), C = Cr is hereditary if and only if r

is hereditary. Now the proof follows by the well known fact that the radical hull of

a hereditary preradical is hereditary as well.

REMARKS 2.6. (a) Concerning (b) in Proposition 2.5 we note that in gene -a1 there

exist non weakly hereditary closure operators C such that @(C) is hereditary (cf.

Example 2.9 below).

(b) Let C be a weakly hereditary closure operator, then for each modul: M and

C-closed submodule N of M a submodule L of N is C-closed in M if and only if it

is C-closed in N (cf. [DG 871).

(c) If C is a maximal closure operator then, for every homomorphism f : M + N, a submodule L of M satisfying L = f-l(f(L)) (or equivalently L I> Keri: 1) is C-

closed if and only if f(L) is C-closed in N, since M/L is isomorphic to Nif (L). For

non-maximal C this is no more true. In fact, if C is idempotent, then the above

property for C yields maximality of C.

For r and s preradicals in R-Mod set

The preradical corresponding to C(r,s) is rC(,,,) = r+s so that C,,, c CI r,s) (this

follows also from Cr+s = Cr + Cs) and consequently C(r,s) = C(r+s,s)) .

Therefore i t is not restrictive to consider only C(r,s) with s < r. Observe: that this

closure operator is maximal if r = s, however the converse is not true (take r cohereditary and s = 0). It follows from Proposition 2.l.(b) that r is idempotent

whenever C(r,s) is weakly hereditary. On the other hand, by Proposiiion 2.5,

C(r,s) is weakly hereditary if s and r are idempotent. It will be shown in Example

2.9 below that idempotency of s is essential.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


PROPOSITION 2.7. Let ( r i : i~ I] and (sj : j~ J ) be families of preradicals. Then

for r = 1 {ri : i s I ) and s = C (s j : j~ J ) the supremum of the family {Cri :

ic I ]u{C% : j~ J ] coincides with C(r,s).

Now we give examples of closure operators which are neither maximal nor

minimal.

EXAMPLE 2.8. For preradicals r , s with s5r it may happen that Cr < C(r,s) < Cr.

Indeed, take R = Z and two distinct sets of prime numbers IP 'cP. Then for

s=tpt, r = t p (see Example 1.3.(b)) we have, (Cr)Q(Z)=Z, (C(r,s))Q(Z)= {a/b :

b is a product of primes in P') and ( C ~ ) Q ( Z ) = (a/b : b is a product of primes in

IP 1. By the above remark C(r,s) is weakly hereditary. A straightforward

verification shows that C(r,s) is also idempotent.

Observe that in the above example for distinct sets IP' contained into a fixed IP we get distinct idempotent weakly hereditary closure operators C(r,s)

corresponding to the radical r. In particular, if IP is infinite, their number is 2w.

EXAMPLE 2.9. Take r as in Example 1.3.(d) with IP = P' the set of all primes.

Then C(t,r) is not weakly hereditary while its preradical t is even hereditary. In

fact, denote by M the C(t,r)-closure of the subgroup Z of Q. Then d(M/Z) = 0

since M/Z = Soc(Q/Z?) is reduced. Thus Z is not C(t,r)-dense in M.

3. Factorization structures

Let E and 7% be classes of homomorphisms in R-Mod such that he E E whenewer e E E and h is an isomorphism, and mh E PL whenever m E PL and h

is an isomorphism. (E, PL) is called factorization structure of R-Mod if: (a) every homomorphism f : M + N can be decomposed as f = me with m E 1Yt

and e E E ((E, Ict)7factorization property);

(b) for each commutative square

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZATIONS, INJECTIVITY AND COMPACTNESS

with e E 2: and m E 7% there is a unique d : N* P such that de = f and md = g

((En)-diagonalization property). For the general theory of factorization smctures

we refer to [Bo 771, [Th 831, [CaHkKe 851, [HeSaSt 871, [DiGi 871.

The best known factorization structure on R-Mod is obtained by taking T t =

(monomorphisms) and Z = (epimorphisms). A less trivial example is obtained by

fixing an idempotent radical r and taking the standard factorization structure defined

by E = (morphisms with r-torsion cokernel] and 7% = (monomorphisn~s with r- torsion-free cokernel). For further examples see Proposition 3.1 below. We note

that there exist factorization structures on R-Mod with 5% non consisting of

monomorphisms (cf.[DiTh 891). In the sequel we consider only factorization

structures with 5% consisting of monomorphisms, or briefly T t c Mono(R-Mod).

For factorization structures (Z,Tt) and ( E ' D ' ) we set (E,Tt) I (2.',Tt1) if T t c 7%'. With this order the conglomerate of all factorization structures of R-

Mod is a complete (illegeatimate) lattice.

Every factorization structure (Z,M) of R-Mod, defines a closure operator C of R-Mod by setting, for each submodule N of an arbitrary module M, Ch4(N) = N',

where N' is the intermediate submodule of the (Z,7%)-factorization of the inclusion

k : N - + M .

Conversely, for each closure operator C, consider the class ZC of all C-dense

homomorphisms and the class TtC of all monomorphisms with C-closed image.

Then R - M o d has the ( ~ C , ~ C ) - d i a g o n a l i z a t i o n property (cf. [IDiCi 87,

Proposition 3.11). Moreover, it was shown in [DiGi 871 and [DiGiTlh 891 in a

more general context that

PROPOSITION 3.1. (a) For each factorization structure (E,N) of R-Mod, the

closure Orator C induced by (Z,Tt) is an idempotent and weakly hereditary

closur~ ~r of R-Mod such that (ZC,Ttc) = (En);

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


(b) For a closure operator C of R-Mod the following conditions are equivalent:

(i) (EC, YLC) is a factorization structure of R-Mod;

(ii) C is weakly hereditary and idempotent.

LEMMA 3.2. Let r be a preradical and f : M -+ N be a homomorphism Then:

(al) f is r-dense if and only if N/f(M) E 7,; (%) f is r-closed if and only if N/f(M) E Fr;

(b,) f is Cr-dense if and only if f(M) + r(N) = N;

(b2) f is Cr-closed if and only if f(M) contains r(N).

PROOF. Follows directly from the definitions.

COROLLARY 3.3. (a) To each factorization structure (@,YL) of R-Mod there corresponds an idempotent preradical r = rC defined by the closure operator C

associated to ( E n ) .

(b) To each idempotent radical r the family of factorization structures ( E n ) ,

having r as associated radical form a complete lattice with smallest element

(ECrnCr ) and largest element (E~',N~').

PROOF. Follows from Proposition 2.5 and Proposition 3.1.

Corollary 3.3 shows that the factorization structures are, roughly speaking,

much more than torsion theories (in Example 2.8 we get, according to Proposition

3.l.(b), 2 W factorization structures with the same radical r). This fact clarifies

Proposition 1.1 from [Fa 881 where only the one-to-one correspondence r -t

( E ~ ~ , Y L ~ ' ) was considered, i.e. only standard factorization structures were

considered. The standard factorization structures (~~',?+l~') where shown to

correspond bijectively to torsion theories also in [DiGi 871, and it was pointed out

that there exist also non-standard factorization structures. In the corollary below we

show that this correspondence is far from being bijective.

Now we characterize the maximal subclass of torsion theories which correspond

bijectively to factorization structures.

COROLLARY 3.4. For a factorization structure (E,n) of R-Mod the following

conditions are equivalent:

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZATIONS, INJECTIVITY, AND COMPACTNESS

(a) the (idempotent) preradical r associated to (E,N) is cohereditary;

(b) there is no other factorization srmcture with the same radical r;

(c) C' = Cr; (d) there exists an idempotent ideal I of R such that r(M) = IM, for each M;

(e) there exists an idempotent ideal I of R such that a homomorphism f : M + N

belongs to TZ if and only i f f is a monomorphism and f(M) 2 IN, and f belongs to

E if and only if f(M)+IN = N.

PROOF. The equivalence of (a) and (d) was remarked in Section 1, (b) is

equivalent to (c) by Corollary 3.3. To show that (e) implies (d) consider M E R- Mod and the (En)-factorization of 0 -t M. According to (e) it is 0 + IN1 -t M. On

the other hand, according to the definition of r (cf. Corollary 3.:i.(a) and

Proposition 3.1), it coincides with 0 + r(M) -t M. Thus IM = r(M). Let us prove

that (a) is equivalent to (c). If r is cohereditary then, for each M E R-Mod and for

each submodule N of M, r(M/N) = (r(M)+N)/N, so by (1) and (3) it fcdlows that (Cr)M(N) = (Cr)M(N). Conversely, if the last identity holds for each N :I M, then

clearly r ns cohereditary by r(M/N) = (r(M)+N)/N which follows as abcve by (1)

and (3). Finally we observe that (b), (c) and (d) imply (e) by Lemma 3.2.(bl) and (b2).

If R admits no non-zero idernpotent ideals, then, by the equivalence of (b) and

(c) in the above corollary, to every non-trivial torsion theory there corr-,spond at

least two different factorization structures.

4. C-injective and absolutely C-closed modules

The relatively injective modules with respect to a (hereditary) torsion theory are

usually studied in torsion theories (cf. [Go 861). Here we prefer to give a slightly

general definition adopting closure operators instead of just torsion theories (i.e.

idempotent weakly hereditary maximal closure operators). To this end we introduce

the notions of injective, absolutely closed and saturated module relaltively to a closure operator C and compare them in Fo(C). We show that these notions

coincide in if C is weakly hereditary and @(C) is a hereditary radical. This

generalizes the known fact in case C is maximal.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

6 2 DIKRANJAN AND GIULI

Let C be a closure operator of R-Mod. A module J is said to be C-injective if for

every module M and for every C-dense submodule N of M each

homomorphism f : N-+ J can be extended to M.

'J (C) will denote the class (full subcategory) of all C-injective modules.

Clearly the usual injective modules coincide with the injective modules relatively

the trivial closure operator (i.e the coarsest closure operator). In case C is a

maximal closure operator generated by a torsion theory r this is the usual definition

of a relatively injective (or r-injective) module (cf. [Go 861). In such a case we

keep the term r-injective instead of Cr-injective. Let us note that the notion of Cr-injectivity coincides with (r,3)-injectivity

defined in [BiKeNe 821.

A great deal of the results known for r-injective modules can be obtained also in

this more general set up.

We begin with the standard facts concerning the existence of C-injective hulls.

If D I C are closure operators then D-density implies C-density, so that C- injective implies D-injective. On the other hand, if D = C* is the weakly hereditary

core of C then C-density is equivalent to C*-density, so that this proves (a) in the

following

LEMMA 4.1. Let C be a closure operator of R-Mod. Then, (a) C-injectivity is equivalent to C*-injectivity and implies C*-injectivity;

(b) 'J(C) is closed under the formation of products; (c) 'J(C) is C*-closed-hereditary;

(d) If M is a C-injective, C-dense submodule of a module N, then M splits off.

PROOF. The proof of (b) and (d) is standard.

(c). By (a) it suffices to show that C-closed submodules of C-injective modules

are C-injective. Let J be a C-injective module and let k : M -+ J be a C-closed

monomorphism. To prove that M is C-injective consider a C-dense submodule N

of a module L and a homomorphism f : N + M. Since J is C-injective then there is an extension f : L -t J of kf. By the continuity property (c3) of closure operators

f (L) is contained in the C-closure of f(N) in J which is contained in the C-closure

of M. Since M is C-closed in J, it follows that f (L) is contained in M. This proves

the C-injectivity of M.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


It will be shown below that under a very natural condition on R, J(C) is closed

also under formation of products (cf. Proposition 4.6).

CoRoLLAKY 4.2. Every essential C-injective submodule is (&-closed. T h ~ s every

C-injective, essential and C*-dense submodule of a module M coincides with M.

PROOF. Let M be an essential C-injective submodule of a module N. Iknote by M' the C*-closure of M in N. Then M is C-injective, and also &-dense and

essential in M'. So, by (d) of the above Lemma, M = M', i. e., M is C*-closed in

N.

If C is a closure operator such that @(C)*=O, then C* is discrete by Proposition

2.1.(~). By Lemma 4.l.(c) every module will be C-injective in this case, being C*-

closed in its injective hull.

For a module M we denote, as usual, by E(M) the injective hull of M. Of course E(M) is CI-injective for each closure operator C. Denote by k ( M ) the Cc-closure

of M in E(M) and by jM : M -+ Ec(M) the inclusion. We show now that, for

idempotent C, Ec(M) is the smallest (up to isomorphism) C-injective module containing M, thus it will be called C-injective hull of M. The reason tc) take C,-

closure instead of just C-closure is that M need not be C-dense in its C-closure if C is not weakly hereditary, while jM is even C*-dense.

PROPOSI'I?ON 4.3. Let C be a closure operator. (a) For each (mono)-morphism f : M + N with N E Ij(C), there exists 3 (mono)-

morphism f : k ( M ) + N such that jMf = f; (b) k ( M ) is C-injective if and only if Ec(M) is C*-closed in E(M). In such a

case it is the smallest C-injective module containing M;

(c) For a module M the following conditions are equivalent:

(i) M is C-injective; (ii) M is C*-closed in E(M);

(iii) M is C*-closed in k ( M ) ;

(iv) M = k ( M ) .

PROOF. (a). By the definition of Ec(M), the monomorphism jM is (:*-dense,

hence C-dense, so that every homomorphism f : M -+ N with N E J ( C ) has an

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


extension f : Ec(M) + N. Moreover M is an essential submodule (of E(M) hence)

of E+(M) so that f is a monomorphism whenever f is a monomorphism.

(b). If Ec(M) is C*-closed in E(M), then by Lemma 4.l.(c), Ec(M) is C-

injective. Now assume that Ec(M) is C-injective. Since it is an essential submodule of E(M), Corollary 4.2 applies. Thus Ec(M) is C*-closed in E(M). The

last part of (b) directly follows from (a). (c). If M is C*-closed in E(M), then it is C-injective by Lemma 4.l.(b) and the

fact that E(M) is C-injective, so that (i) implies (ii). Conversely assume M C- injective. Since M is a C*-dense essential submodule of Ec(M), it follows from

Corollary 4.2 that M = Ec(M). Now (b) implies that M is C*-closed in E(M).

Clearly (ii), (iii) and (iv) are equivalent.

By (b) EC(M) is C-injective whenever C* is idempotent, in particular, whenever

C is idempotent.

The following proposition provides examples and ~ounterexamples related to C-

injective modules.

PROPOSITION 4.4. (a) For an idempotent radical r the following conditions are

equivalent:

(i) r is hereditary;

(ii) every M E Fr is C,-injective.

In particular, if r is an idempotent, cohereditary radical then every M E f, is r-

injective if and only if r is hereditary.

(b) For a hereditary radical r the following conditions are equivalent:

(i) r is stable, i.e. 7, is closed with respect to injective hulls; (ii) C,-injective is equivalent to injective in 7,;

(c) Let r be a preradical; then r-injectivity implies injectivity in the following

cases:

(1) r contains z;

(2) R is a left hereditary ring (= quotient of injective modules are injective) and

every injective module is r-torsion.

PROOF. (a). It is known that r is hereditary if and only if E(M) is r-torsion-free for each M E f r. Now C, = (C,)* by Proposition 2.5.(b), so that, according to

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZATIONS, INJECTIVITY AND COMPACTNESS 6 5

Proposition 4.3.(c), M is Cr-injective if and only if M is Cr-closed in E(M), which

is equivalent to r(E(M)) c M. If M E Fr then clearly E(M) E Fr. (b). If r is hereditary, by the above remark M is Cr-injective if and only if

r(E(M)) c M.

(i) => (ii). If M E Tr, then by (i) E(M) E 7,. If M is Cr-injective then E(M) =

r(E(M)) c M . (ii) => (i). Assume that M E Tr and set M1 = r(E(M)). Then M1 = (Cr),yM)(M),

since M c MI. Hence M1 is C,-injective and r-torsion. By (ii) M1 is injective, thus

MI = E(MI) = E(M). SO E(M) E 7,.

(c).(l). Now each module M is r-dense in its injective hull E(M) and M is r -

injective if and only if M is r-injective in the usual sense

(c).(2). Let M be a r-injective module. Then M is r-closed in E(M). On the other

hand E(M)/M is injective since R is left hereditary. Hence E(M)/M E T r ,

consequently M is r-dense in E(M). Thus M = E(M) is injective.

As the following example shows C-injectivity strongly depends not only on the radical r = rC but also on the concrete choice of the operator C with radical r .

EXAMPLES 4.5. (a) Let r be a hereditary radical. Then by Proposition 4.4.(a), every M E Fr is Cr-injective. On the other hand if r contains z then by Prlsposition

4.4.(c) the r-injective modules are precisely the injective modules, so :hat C,-

injective does not imply r-injective.

(b) It is well known that if r is a hereditary radical then a module J is r.lnjective

if and only if for each r-dense left ideal I of R, each homomorphism f : I -+ J can be

extended to R (see, e.g.[Go 86, Proposition 8.21). For non maximzl (even

hereditary) closure operators C the above result is false. In fact take R = Z! and C =

C,. Then, z is stable, so by Proposition 4.4.(b) z-injectivity coincides with

injectivity for z-torsion modules. On the other hand the unique C,-dense ideal of Z

is Z , so that every group satisfies this weaker version of z-injectivity.

Let C be a closure operator of R-Mod. We say that R is C-hereditaly if every C-

dense left ideal of R is projective. Clearly every left hereditary ring is C-hmereditary.

PROPOSITION 4.6. Let C be a maximal hereditary closure operator. If R is C-

hereditary then the quotient of every C-injective module is again C-injective, i.e.

J (C) is closed under quotients.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


PROOF. Let M be a C-injective module and N be a submodule of M. To show that

M/N is C-injective it suffices to prove that for each C-dense left ideal I of R every

homomorphism f : I + M/N can be extended to R (see (b) of the above example).

Now, by the C-projectivity of R, the ideal I is projective, so that there exists a

homomorphism g : I + M such that for the quotient homomorphism q : M + M/N,

qg = f. Now by the C-injectivity of M there exists a homomorphism r : R + M

extending g. Then the composition qr extends f.

Let C be a closure operator of R-Mod. A module M E fO(,-) is said to be

absolutely C-closed (respectively: C-saturated) if every monomorphism f : M + N with N E f is C-closed (respectively: an isomorphism).

In case C = C' for a hereditary radical r , absolute C-closedness coincides with

absolute r-purity defined by Freyd [Fr 641 (see also [Go 861). Also in this case

we keep the term absolutely r-closed (r-saturated) for absolutely Cr-closed (Cr-

saturated).

By definition absolute C-closedness implies C-saturatedness and the converse is

true whenever C is weakly hereditary in f,. On the other hand, if r = @(C), then

absolute r-closedness implies absolute C-closedness and analogously for

saturatedness. The converse is not true since, for example, every r-torsion-free

module is absolutely Cr-closed for each preradical r. Clearly absolute C-closedness coincides with absolute C*-closedness. If C* coincides with the discrete closure

operator on every @(C)-torsion-free module then every @(C)-torsion-free module

is C-saturated.

The result below collects the relations between C-injectivity, C-saturatedness and

absolute C-closedness.

PROPOSITION 4.7. Let C be a closure operator and let r = @(C). For M E f, the

following conditions are equivalent:

(i) M is C-injective;

(ii) M is C-saturated and Ec(M) E f, . In particular if r is hereditary and C is weakly hereditary, then a module M E Fr is

C-injective if and only if it is C-saturated if and only if it is absolutely C-closed.

PROOF. (i) => (ii). If M is C-injective, then clearly EC(M) = M E f r . To show

that M is C-saturated consider a module N E f, and a C-dense monomorphism

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


f : M + N. Then, by Lemma 4.l.(b) there is a submodule N' of N such t~hat N =

N' O f(M). Then N' E F, is a submodule of N. On the other hand the C-density

of f(M) in N implies the CT-density of f(M) in N. Thus, in virtue of Lemma 3.2.(al), N' .r N/f(M) is r-torsion. Therefore N' = 0, i.e. f(M) = N.

(ii) => (i). Let M E fr and let M be C-saturated. Since the inclusion of M in Ec(M) is a (C,-dense, hence) C-dense monomorphism, then it is an isomc~rphism

whenever E(:(M) E F,. By Proposition 4.3.(c), M = Ec(M) is C-injective.

To finish the proof assume r hereditary and M E F,. Then E(M) t F,,

consequently EC(M) E f Thus C-injectivity and C-saturatedness for M are

equivalent by the above argument. Since C is weakly hereditary, the) imply

absolute C-closedness.

REMARKS 4.8. (a) In the last statement the hereditarity of r is essential. In fict, let

r be a cohereditary and non-hereditary radical. Then by Corollary 3.4 eve]?{ M in

F, is absolutely r-closed since, in F,, Cr coincides with the discrete c osure

operator. On the other hand by Proposition 4.4.(a) there exists a non r-injective

module M E F,.

(b) If C is a weakly hereditary closure operator then, as observed befcre, C-

saturatedness and absolute C-closedness coincide. Thus, by Proposition 2 6, for

weakly hereditary C, every C-injective module is absolutely C-closed. Th- next

example shows that the converse need not be true, even for a maximal and weakly

hereditary closure operator C (although, according to Proposition 4.7, it is true for

arbitrary closure operators C with hereditary @(C)). (c) For r a:; in example 1.3.(d) every abelinn group is r-injective, since r*= 0. On

the other hand the group Z is not absolutely r-closed (see Example 2.9 and take

into account that r(Q)=O).

(d) As mentioned at the end of the above proof, if r is hereditary, then C-

injectivity coincides with C-saturatedness, so that both properties are implied by

absolute C-closedness in this case. If C is not weakly hereditary they do not

coincide with absolute C-closedness. In fact, take r as in (c) and C=C(t,r). Then @(C)=t and (1 coincides with Cr on every torsion-free group. Thus C is discrete on

every torsion-free group. Hence every torsion-free group is C-injective, while Z is

not absolutely C-injective. In fact, Z is not C-closed in Q (see (c)).

EXAMPLE 4.9. Let p be a prime number. Then the group G of p-adic integers is

absolutely d-closed. For a proof of this fact see Theorem 5.22 below. On the other

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

68 D I K R A N J A N A N D G I U L I

hand G is not d-injective since by Proposition 4.4.(~).(2) every d-injective group

is injective, i.e. divisible.

5. C-compact modules

Let C, D be closure operators of R-Mod. A homomorphism f : M + N in R- Mod is called (c,D)-closed if for each C-closed submodule L of M, f(L) is a D-

closed submodule of N. If C = D we speak of C-closed homomorphisms. Clearly,

composition of C-closed homomorphisms is C-closed. Conversely, if a

composition gf of two homomorphisms is C-closed and f is surjective (g is injective) then g is C-closed (f is C-closed). If C = v{Ci: i E I ) and D = v{Di) : i

E I) , then a homomorphism is (C,D)-closed whenever it is (Ci,Di)-closed for each

i E I. In the next lemma we give some useful properties of the C-closed

homomorphisms.

LEMMA 5.1. Let s I r be preradicals, C=C(s,r) and f : M + N be a

homomorphism. Then:

(a) f is Cr-closed if and only if f(r(M))=r(N);

(b) r is cohereditary if and only if each homomorphism is Cr-closed;

(c) r is a radical if and only if for each module L the quotient homomorphism

L -+ L/r(L) is Cr-closed;

(d) iff is surjective and Ker f c r(M), then f is Cr-closed;

(e) i f f is surjective, r is a radical and Ker f c r(M), then f is Cr-closed and C-

closed.

PROOF. A straightforward verification gives (a), which implies (b) and (c). To

prove (d) observe that every r-closed submodule of M contains r(M), so that

Remark 2.6.(c) applies. Finally, the first part of (e) follows from (a) and

Proposition l.l.(d). To prove the second part of (e) take a C-closed submodule A

of M. Then by Lemma 3.2.(b2) A 2 r(M) 1 Ker f, so by Remark 2.6.(c) f(A) is

s-closed in N since A is s-closed in M. This proves that f is (C,Cs)-closed. By the first part f is C,-closed, thus f is C closed since C=Cr v @.

Observe that in the hypothesis of the above lemma a surjective homomorphism f:M + N with Ker f c r(M) is C-closed provided s = r or r is a radical.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZATIONS, INJECTIVITY, AND COMPACTNESS 69

Let X be a non empty class of modules. A module M is called (C,D)-compact

with respect to X if the projection p : MxN + N is (C,D)-closed for each r\l E Y. In case 4: = R-Mod we speak of (C,D)-compact modules. If, C = D we j3eak of

C-compact modules with respect to L a n d if in addition 1 = R-Mod we speak of

C-compact modules. Following [Fa 881 we abreviate Cr-compact to r-cony7act.

Clearly M is (C,D)-compact if and only if M is (C*,D*)-compact. I f f : Wi + N is

(C,D)-closed and g: N -+ P is (D,E)-closed, then gf is (C,E)-closed.

The following result is standard for compactness in topology.

PROPOSITION 5.2. Let C and D be closure operators of R-Mod and X be a class

of modules. If M is (C,D)-compact with respect to and M' is a submodule of M

then M/M' is (C,D)-compact with respect to X.

PROOF. Denote by q : M -+ M/M' the quotient homomorphism and let N be an

arbitrary module in 1 and p : M/M'xN + N be the projection. Let A be a C'-closed

submodule of (M/Mt)xN. Then A1 = ( q x l N ) - l ( ~ ) is C-closed in MxN. Dmenote by

p' the projection MxN + N, then p(qxlN) = p'. Thus p(A) = p((qxlN:)l:A1)) =

pl(Al) is D-closed in N.

It follows from the proposition above that there exist (C,D)-compact modules if

and only if C* I D * . In fact we have

PROPOSITION 5.3. For closure operators C and D the following conditions are

equivalent:

(i) There exists a (C,D)-compact module;

(ii) The module 0 is (C,D)-compact; (iii) D* I C*.

PROOF. (i) is equivalent to (ii) by Proposition 5.2 and the latter is equivalent to (iii)

by definition of (C,D)-compactness.

PROPOSITION 5.4. Let C and D be closure operators, with C* 2 D*, and X be a

class of modules.

(a) If M is a (C, D)-compact module with respect to X, then:

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

DIKRANJAN AND GIULI

(I) M is (C,D')-compact with respect to X,for each D' I D;

(2) M is (C',D)-compact with respect to 1 , for each C I C'. (b) Every @(C)-torsion module is (C,D)-compact.

(c) If C and D are minimal, then every module is (C,D)-compact. (d) If C = v{Ci: i E I ) and D = v{Di) : i E I ) , then a module M is (C,D)-

compact with respect to y whenever it is (Ci,Di)-compact (with respect to 1) for each i E I.

(e) For an arbitrary preradical r, (C,D)-compactness with respect to 1 implies

(CvCr,DvCr)-compactness with respect to 1. (f) If C = C(r,s), with s I r , then a module is C-compact if and only if it is

(C,Cs)-compact.

PROOF. (a). Trivial.

(b). Let M be a @(C)-torsion module. Then, for each module N, every C-closed

submodule L of MxN contains M, so that L = MxL' for L' = LnN. Therefore L'

is a C-closed submodule of N. This proves that M is C-compact. By (a), this

implies that M is (C,D)-compact.

(c). Set r = @(C). A submodule of a module K is C,-closed if and only if it

contains r(K). Let now M and N be arbitrary modules and A be a C-closed

submodule of MxN. Then A contains r(MxN) = r(M)xr(N). Thus the projection p

: MxN + N is C-closed. This proves that M is C-compact. By (a), this implies that

M is (C,D)-compact.

(d). It suffices to observe that a submodule is C-closed (resp. D-closed) if and only if it is Ci-closed (resp. Di-closed) for every i E I.

(e). Follows directly from (c) and (d).

(f). If a module M is (C,C)-compact (= C-compact) then by G I C and (a).(l),

M is (C,Cs)-compact. Conversely, by (e), (C,Cs)-compactness implies (CvCr,CSvCr)-compactness and the latter coincides with C-compactness.

In (a) of the next proposition we show that the C-compactness with respect to a maximal closure operator C can be studied within the category Fr. In (b) we

establish an analogue of Lemma 4.2 (c) for CT-compactness.

PROPOSITION 5.5. Let r and s be preradicals, C=C(r,s), M be a module and X be

a class of modules.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZATIONS, INJECTIVITY AND COMPACTNESS 7 1

(a) If r is a radical or s=r, then M is C-compact with respect to X if and only if

M/r(M) is C-compact with respect to Y. (b) If r and s are idempotent, then C-compactness with respect to 41 is C'. closed

hereditary.

PROOF. (a). One implication follows from Proposition 5.2.(a). Assume that M/r(M) is C-compact with respect to Y. Let A c MxN, N E Y, be C-closd; then

r(M) c A. Consider the quotient homomorphism q : M -+ M/r(M) and set p ' = q X l ~ . Then A I> Ker(p') so that the homomorphism p' is C-closed by Ixmma

5.l.(d) and (e). The projection p": M/r(M)xN + N is C-closed by the compiictness

of M/r(M). Now the projection p :MXN -+ N is the composition of p' anti p" so

that p is C-closed.

(b). Let M be a C-compact module and L be a C-closed submodule of I f . Take

an arbitrary N in Y and a C-closed submodule K of LXN. Then K is C-c1osl:d also

in MXN since LXN is C-closed in MXN and C-closedness is transitive by the weak

hereditarity of C (cf. Remark 2.6.(b)). In fact, the idempotency of r and s yields

the weak hereditarity of C by Proposition 2.5.(b) and the definition of C.'I'llus the

projection of K into N is C-closed.

In the case of an idempotent radical r = s, property (b) was given alscl in [Fa

88, Proposition 4.41.

PROPOSITION 5.6. Let C be a closure operator of R-Mod and let r be a preradical.

If M is a (Cr,C)-compact module then for each homomorphism f : M -+ N, f(M) is C-closed in N whenever r*(N) c f(M).

PROOF. Set Q = N/r*(N) and consider the projection p : MxQ + Q wnich is

(Cr,C)-closed by the (Cr,C)-compactness of M. Consider now h = qf where q:

N -+ Q is the quotient homomorphism. To prove that f(M) is C-closed in N it is enough to show that h(M) is C-closed in Q. In fact, by r*(N) = ker q c f(M) it

follows that f(M) coincides with q-l(h(M)). Denote by D the graph of the

homomorphism h, i.e. D = ((x,h(x)): x E M). If we consider Q as a subrnodule

of MxQ in the obvious way then Q interscts D in 0 and Q+D = MxQ. Therefore

(MxQ)/D is isomorphic to Q, hence it belongs to Fr by the definition of Q. 'Thus D

is an r-closed submodule of MxQ by Lemma 3.2.(a2). Then p(D) = h(M) is C-

closed in Q.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


COROLLARY 5.7. For every (C,D)-compact module M and each homomorphism f:M 4 N with N E fqCY the submodule f(M) is D-closed. In particular, every C-

compact module in f qC) is absolutely C-closed.

PROOF. Set r =Q(C), then CICr , thus M is (Cr,D)-compact by Lemma 5.3.(a).(2). If N E fQ(C), then r*(N) = 0 c f(M) so that f(M) is D-closed in N by

Proposition 5.5.

COROLLARY 5.8. If C is a maximal closure operator then a module M is C-

compact whenever there exists a C-dense and C-compact submodule N of M.

PROOF. Let r = Q(C). By (a) of Proposition 5.5 it suffices to show that M/r(M)

is C-compact. Denote by N' the image of N in M/r(M). Then N' is r-compact by

Proposition 5.2 and N' is r-dense in M/r(M). Since M/r(M) E fr, Corollary 5.7

implies that N' is also r-closed in M/r(M), thus M/r(M) = N' is r-compact.

In case C is also hereditary and idempotent (i. e. Q(C) generates a hereditary

torsion theory), the above result was obtained also in [Fa 88,Proposition 4.51 .

Observe that Proposition 5.5.(a) and Corollary 5.8 are particular cases of the

general problem: is M C-compact provided a submodule N of M and the quotient

M/N are C-compact ? Another case of this problem will appear in Lemma 5.17.

PROPOSITION 5.9. Let r and s be preradicals and C = C(r,s). If for every C-

closed submodule N of a module M the quotient M/N is absolutely s-closed then M

is C-compact.

PROOF. Consider an arbitrary module N and the projection p : MXN + N. Let A be a C-closed submodule of MxN. Then A contains r(MxN) = r(M)xr(N). Therefore

p(A) contains r(N), consequently the C-closedness of p(A) in N is equivalent to

the fact that N/p(A) is s-torsion-free. Since M+A = M+p(A), this quotient is isomorphic to the quotient of N'= (MxN)/A with respect to its submodule M' =

(M+A)/A, so it suffices to show that the module N'/M' is s-torsion-free. By the

definition of CS this will follow from the fact that M' is s-closed in N'. Now

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


If s = r (i.e. C is maximal) then r-closed, s-closed and C-closed submodules of

M coincide, hence (0 => (a) by Proposition 5.9.

It will be shown in Example 5.24 that the first implication is not reversible even

for idempotent radicals r and s. The case of a maximal closure operator will be

given separately in w e of its importance.

COROLLARY 5.1 1. Let r be a preradical. Then a module M is r-compact if and

only if for each r-closed submodule N of M the quotient M/N is absolutely r -

closed.

C-compactness of M is available under stronger conditions involving C-

injectivity as explained in the following corollary. In case r = s they coincide.

COROLLARY 5.12. Let r and s be idempotent preradicals with s I r and set C =

C(r,s). For a module M consider the following conditions:

(a) M is (Cr,Cs)-compact;

(b) M is C-compact;

(c) M/N is C-injective for each C-closed submodule N of M.

(d) M/N is C-injective for each r-closed submodule N of M.

Then (c)=>(b)=>(a)<=(d) <= (c). If r is hereditary then (a) and (d) are equivalent.

PROOF. (b) => (a). Follows from Lemma 5.3.(a).

(c) => (d). Follows from the fact that r-closed implies C-closed.

(d) => (a). By Proposition 2.5.(b) and the idempotency of r and s, C(r,s) is

weakly hereditary. Therefore C(r,s)-injectivity implies absolute C(r,s)-closedness

for every module in F,, according to Remark 4.8.(b). This proves the implication

by the equivalence of (c) and (e) in Theorem 5.10.

Anagously s-injectivity implies absolute s-closedness for every module in F,. Since every C-closed module is s-closed and C-injectivity implies s-injectivity,

then the implication (c) => (b) follows from Proposition 5.9.

On the other hand, according to Proposition 4.7 every absolutely r-closed module

is r-injective whenever r is hereditary. Now Theorem 5.10 can be applied again to

get (a) => (d).

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

F A C T O R I Z A T I O N S , I N J E C T I V I T Y AND C O M P A C T N E S S

We give separately the case of a maximal idempotent closure operator.

COROLLARY 5.13. ([Fa 88, Theorem 2.31) Assume r idempotent radical and let

M be a module.If for every r-closed submodule N of M the quotient h4/N is r -

injective, then M is r-compact. If r is hereditary, then the converse holds.

If every quotient M/N of M such that M/N is absolutely C-closed, satisfies also

E(M/N) E F,, then the above equivalence is available without the cordition of

hereditariness.This follows from Proposition 4.7 since now every quotient M/N

which is absolutely C-closed is also C-injective.

COROLLARY 5.14. Let r be a hereditary radical such that R is C-hereditmy for C =

Cr. Then a module M is r-compact if the quotient M/r(M) is r-injective.

In particular a module M E Fr is r-compact if and only if it is r-injective.

PROOF. Suppose that the quotient M/r(M) is r-injective. Let now N be ar arbitrary

r-closed submodule of M. Then N contains r(M), so that the quotient M/N is a

quotient of the the r-injective module M/r(M). By the C-hereditarity of R: it follows

from Proposition 4.6 that also M/N is r-injective. By Corollary 5.13 M r-sompact.

It follows from Corollary 5.13 that all conditions in Corollary 5.12 are

equivalent provided r = s is hereditary, i.e. C is a maximal hereditary closure

operator. In the following example we show that in general one can not hope to

prove that C-compactness yields the condition (c) in Corollary 5.12 even if C is

hereditary.

EXAMPLE 5.15. Take R = 2 and t as in Example 1.3.(a). By Proposititm 5.4.(c) every module is Ct-compact. Take now M = Z and N = pZ. Then N is. Ct-closed

in M while M/N = Z(p) is not Ct-injective since the group Z(p) is not Ct-injectve

for any prime p by Proposition 4.4 (b).

The closure operator considered in the above example is hereditsuy but not

maximal. We consider now another example in which the closure operator is

maximal.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

76 D I K R A N J A N AND GIULI

EXAMPLE 5.16. Take R = Z and n E W. Then for C = C" (= C, by Corollary 3.4)

every module is C-compact by Proposition 5.2 (d). On the other hand each non-

divisible n-torsion free group M is not C-injective, by Proposition 4.4.(~).(2),

while (0) is a C-closed submodule of M.

Notice that in the example above the closure operator is not weakly hereditary, i.

e., n is not idempotent. For an example in this direction of a closure operator

which is both maximal and weakly hereditary (as well as idempotent) see Theorem

5.22 below.

Now we consider products of C-compact modules. It is known in a very general

context that C-compactness is finitely productive (cf. [Ma 741, [Ca 891, [DiCi 891). In what follows we shall show that C-compactness is productive whenever C

= Cr for a hereditary Jansian radical r such that R is Cr-hereditary (r is a Jansian

radical if it commutes with products).

First we prove a lemma contained in [Fa 88,Theorem 3.21 in case R is r -

hereditary. The proof we give here is based on the same idea.

LEMMA.5.17. If r is a hereditary radical, N is an r-compact submodule of M and

M/N is r-compact, then M is r-compact.

PROOF. According to Corollary 5.13 it suffice to show that for each r-closed

submodule L of M the quotient M/L is r-injective. Clearly L n N is an r-closed

submodule of N, so by Corollary 5.13 the quotient N/(NnL) is r-injective. Thus it

is r-closed considered as a submodule of M/L since M/L is r-torsion free and, according to Proposition 4.7, L/(LnN) is absolutely r-closed. Hence the quotient

of M/L with respect to this submodule is r-torsion free. It is isomorphic to

M/(N+L), so to the quotient of M/N with respect to its submodule MI(LnN). By

the r-compactness of M/N and Corollary 5.13, the quotient M/(N+L) is r-injective

since the submodule (L+N)/N of M , , is r-closed by Remark 2.6.(c). Now M/L has an r-injective submodule (N/(NnL)) and the quotient with respect to this

submodule (E M/(N+L)) is r-injective. Then also M/L is r-injective (cf. [La 711).

By Corollary 5.14 and Lemma 4.1 .(b) the product of r-compact r-torsion-free

modules is r-compact for R and C as in Corollary 5.13.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

FACTORIZATIONS, INJECTIVITY, AND COMPACTNESS 7 7

Recall that, according to Proposition 5.4.(b), every r-torsion module is r -

compact. In the next corollary we show that the productivity for r-com~pactness is

determined by means of r-torsion modules.

COROLLARY 5.18. For a hereditary radical r and a family (Mi : i (E I ) of r -

compact modules consider the following conditions:

(a) the product M = n { M i : i E I ) is r-compact;

(b) the product N = n(r(Mi) : i E I ) is r-compact.

Always (a) => (b).

If R is Cr-hereditary, then the class of r-compact r-torsion-free modules is closed

under products and (a) <=> (b).

PROOF. The first implication follows from Proposition 5.5.(b) since PJ is an r -

closed submodule of M. Suppose that (b) holds and R is Cr-hereditary. According

to Lemma 5.17, to prove (a) it suffices to show that the quotient P4/N is r -

compact. It is isomorphic to the product of the r-injective modules Mi/r(Mi).

Therefore it is r-injective by Lemma 4.l.(b). By Corollary 5.14 M/N is r-compact

since M/N is r-torsion-free.

Now we give an example of product of r-compact modules which is not r -

compact (i.e., the additional condition (r-torsion-free) in Corollary 5.18 is

essential). According to Corollary 5.18 it is enough to consider only torsion

modules.

EXAMPLE 5.19. Let R = iZ and p be a fixed prime number. Take now M, =

Z/pnZ, for n E M. Then the product M = n(M, : n E N) is not t-cornpact. In

fact M/t(M) is not divisible, so by Lemma 4.5 it is not t-injective. By Corollary

5.13 M is not t-compact. Since in this case t is a hereditary radical, this corrects an

erroneous statement in [Fa 881 p. 1218.

In what follows we show that for r-compactness the class of "test mcdules" can

be restricted to the class of r-torsion-free modules. On the other hand the example

which follows shows that r-compactness with respect to { R ) is in general weaker

than r-compactness.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


PROPOSInON 5.20. Let r be a radical and s be a preradical satisfying s 2 r . Then

for C=C(r,s) C-compactness coincides with C-compactness with respect to F,.

PROOF. Suppose that a module M is C-compact with respect to F,. For an arbitrary

module N consider the projection p : MxN + N. To show that p is C-closed take a

C-closed submodule A of MxN. Then A contains r(MXN). From now on we

identify N with the submodule ( 0 ) xN of MxN. In these terms A I> r (N) .

Consider now the commutative diagram

MxN - N

where p' is the projection, q is the quotient homomorphism and f = lMXq. Then

r(MXN) 2 Ker f, so that, according to Lemma 5.l.(b) f is C-closed. Since N/r(N)

is r-torsion-free, then the C-compactness of M yields that p' is C-closed. Then

qp=p'f is C-closed, so that q(p(A)) is C-closed in N/r(N). Thus p(A) = q- I(p'(f(A))) (by p(A) 2 Ker q, since A 3 r(A)) is C-closed in N.

It follows from Proposition 5.4.(a) and the above proposition that in the above

hypothesis C-compactness coincides with s-compactness with respect to the class

f ,. Observe that by Proposition 5.4.(a) s-compactness coincides with s-

compactness with respect to the larger class 7',, and C-compactness does not imply

s-compactness even for idempotent radicals r and s (see Example 5.24 below).

EXAMPLE 5.21. Every abelian group is d-compact with respect to ( R ) , since all

cyclic groups are reduced. On the other hand we shall show now that an abelian

group is t-compact with respect to { R ) if and only if it is divisible modulo t.

According to Proposition 5.5.(a) it suffices to show that every torsion-free group M which is t-compact with respect to { R ] is divisible. In fact, let x E E(M) and N

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

F A C T O R I Z A T I O N S , I N J E C T I V I T Y AND C O M P A C T N E S S 7 9

be the cyclic soubgroup of E(M) generated by x. Then N is isomorphic to R, thus the projection p : MxN -+ N is t-closed. Now consider the subgroup D =. ((x,x) : x

E M n N ) of MxN. The submodule D is t-closed in MxN, thus p(D) = ILInN is t-

closed in N. Since M c E(M) is essential, it follows that M n N # { O ) , so that

M n N is a t-closed submodule of N if and only if M n N = N, i.e. M c N.

Therefore M = E(M).

It will be interesting to characterize the C-compactness with respect to the class

of all cyclic modules or with respect to the class of finitely generated mocldes.

We conclude the paper with a result which connects C-compactness with the

classical algebraic compactness (cf. [Ka 541, [Or 791). It provides a nm-trivial

example of r-compactness in the case r is not hereditary, while the con3tion (c)

considered in Corollary 5.12 gives injectivity by Lemma 4.5. This stlows that

Corollary 5.13 cannot be extended to non-hereditary radicals.

Recall that a group M is cotorsion if Ext(Q,M) = 0, so that every algebraic

compact goup is cotorsion.

THEOREM 5.22. Every algebraically compact abelian group is d-compact

PROOF. L,et G be an algebraically compact abelian group. According to Theorem 1.5 from [Or 791 and Proposition 3.3.(a2), for every d-closed subgroup H of G

the quotient GIH is reduced and algebraically compact. To prove the t ieorem it

suffices to show, by Corollary 5.1 1, that every reduced algebraically compact

abelian group is absolutely d-closed. Assume G reduced; then by Corollary 5, p. 190 from [Or 791, G = n {Gp : p prime) , where each group G is conplete in P the p-adic topology. Now suppose that N c G and NIG is not reduced, i.e. G is

not d-closed in N. We shall show that also N is not reduced. We can assume

without loss of generality that N/G is isomorphic to either Q or the Priifer group

Z(pm) for some prime p.

In the first case G being cotorsion splits in N, thus N is not reduced.

Assume now that N/G is isomorphic to Z(pM). Set G' = n ( G q : q i t p); then

G' is p-divisible since every Gq is p-divisible. Moreover G = GtxGp. Let (h',)

be a sequence of elements of N, such that the family (htn+G) is a canonical

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

80 DIKRANJAN AND G I U L I

system of generators of N/G z Z(p) , i.e. P ~ ' , + ~ E htn+G for each n E M and

ph', E G. Let pYl = g ' l + ~ l , g'] E G , x1 E Gp . Then g'] = pgl with gl E G',

since G' is p-divisible, so that p(Yl-gl) = xl. Set hl = Wl-gl; then (h-Kl) E G,

so they give the same coset in NIG and phl = xl . Let phI2 = h1+gt2 = x2 with gI2

E G', x2 E Gp. Then gI2 = pg2, g2 E G' and for h2 = hI2-g2 we have ph2 =

h +x2

Assume that for a natural n we have found h, = 0, hl , ... , h , ~ N, x l , ..., x, E

Gp, such that

(4) phk+] = hk+xk+l for 0 I k < n and h,-h', E G.

Now this gives ~ h ' , , ~ = hn+g'n+l+x,+l, for some x ,+~ E Gp , g'n+l E G'. By

the p-divisibility of G' there exists gn+l E G' with g'n+l = pgn+~ . Setting hn+l =

h'n+l-gn+l we get ph,+, = hn+xn+]. So we defined a sequence (h,) of elements

of N, satisfying (4) for each k E M. Consider now the subgroup Gp of N and the quotient N/Gp; both they are q-

divisible for each prime q # p, so the group N has the same property. By the completeness of Gp in the p-adic topology the elements ak = C(pmxk+, : m =

0,1, ...) of Gp are uniquely determined and satisfy

Now (4) and (5) give

If the element hl-al of N is non-torsion, then (6) and the fact that N is q-divisible

for each q # p imply that N contains a copy of Q. Hence N is not reduced. If hl-al

is torsion, then it is in fact p-torsion, since N/Gp is p-torsion and t(Gp) = tp(Gp) by

the completeness of Gp with respect to the p-adic topology. Thus, by (6), the

subgroup of N generated by (h,) is isomorphic to Z(pm). This implies again that

N is not reduced.

The above argument repeats a substantial part of a similar argument given in [Or 791, p. 194, Lemma.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


Denote by o the intersection of the radicals n . Then o* = d, while F, consists

of all groups M such that the natural topology of M (having as :I base of

neighbourhoods of zero the family of subgroups {nM : n = 1,2, ...)) is Heusdorff.

COROLLARY 5.23. Every cotorsion group is d-compact. Conversely, svery d-

compact group G satisfying o(G) = 0 is algebraically compact (hence

PROOF. According to Proposition 5.5.(a) it suffices to show that every reduced

cotorsion group is d-compact. Let M be a reduced cotorsion group. Thm M is a

Z-module, where Z is the completion of Z with respect to the natural top'alogy [Ro 681. There exists a homomorphism f from the direct sum Z(&) of cr copies of Z onto the group M. Since the quotient zOL/z(OL) is torsion-free, f can be e:c:ended to fl : ZOL_t M. Since Za is algebraically compact, it follows from Theorem 5.22 that

ZOL is d-compact. Now by Proposition 5.2 M is also d-compact.

Now assume that G is a d-compact group satisfying o(G) = 0. Denote by G' the

completion of G in the natural topology. Then G' is a reduced group an'rl G1/G is

divisible, i.e. d-dense.On the other hand G is d-compact and reduced, so by

Corollary 5.11 the group G is absolutely d-closed, thus G = G'. Being complete

in the natural topology, the group G is algebraically compact (Theorem 2.4, from

[Or 791, p. 212 )

Notice that the d-injectivity coincides with injectivity (=divisibility) by

Proposition 4.4.(~).(2). Therefore in this case, an abelian group M sat sfies the

condition (c) in Corollary 5.12 if and only if M is divisible. On the other hand

Theorem 5.22 provides a large class of examples of d-compact groups which are

not divisible (so do not satisfy (c) in Corollary 5.12).

EXAMPLI: 5.24. Let p be a prime number, set r = d p and s=d (see Example

1.3.(c)). Now the group M = Zq (the localization of Z at q) is p-divisible, hence

r-torsion and consequently C-compact for C=C(r,s). On the other hand the natural

topology of M is Hausdorff, i.e. o(M)=O, and M is not algebraically compact.

Thus by the above corollary M is not s-compact.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

REFERENCES

DIKRANJAN AND GIULI

[BePost 701 M.P. Bem, J.R. Porter and R.M. Stephenson, A survey of minimal topological spaces, Proc. Kanpur Topological ~onfkrence (~cademicPress, 1970) 93- 114.

[BiKeNe 821 L. Bican, T. Kepka and P. Nemec, Rings, Modules, and Preradicals, Lecture Notes in Pure and Applied Math. 75 (Marcel Dekker, New York, 1982).

[Bo 771 A.K. Bousfield, Constructions of factorization systems in categories, J. Pure Appl. Algebra 9 (1977) 207-220.

[BrGiHe 891 G.C.L. Briimmer, E. Giuli and H. Herrlich, Epireflections which are completions. Preprint.

[CaHeKe 19851 C. Cassidy, M. HCbert and G.M. Kelly, Reflective subcategories, localizations, and factorization systems, J. Austral. Math. Soc. (Ser. A) 38 (1985) 387-429.

[Ca 891 G. Castellini, Compactness, surjectivity of epimorphisms and compactifications. Preprint.

[DiGi 871 D. Dikranjan and E. Giuli, Closure operators I, Topology Appl. 27 (1987) 129-143.

[DiGi 88a] D. Dikranjan and E. Giuli, S(n)-@-closed spaces, Topology Appl. 28 (1988) 59-74.

[DiGi 88b] D. Dikranjan and E. Giuli, Compactness, minimality and closedness with respect to a closure operator, Proceedings of the International Conference on Categorical Topology, Prague, 1988 (World Scientific Publ., Singapore 1989).

[DiGi 891 D. Dikranjan and E. Giuli, On C-perfect maps and C-compactness. Preprint.

[DiGi 901 D. Dikranjan and E. Giuli, Epimorphisms and completions in categories of modules, Manuscript.

[DiGiTh 891 D. Dikranjan, E. Giuli and W. Tholen, Closure operators 11. Proceedings of the International Conference on Categorical Topology, Prague, 1988 (World Scientific Publ., Singapore 1989).

[DiCiTh 901 D. Dikranjan, E. Giuli and W. Tholen, Closure operators in topology and algebra. Preprint.

[DiTh 891 D. Dikranjan and W. Tholen, Concrete and abstract views of factorization systems, Manuscript.

[Fa 881 T. Fay, Compact Modules, Comm. in Algebra 16(6) (1988) 1209-1219.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13


[FaWa 89.1 T. Fay and G. Walls, Compact Nilpotent Groups, Comm. in Algebra, 17(9) (1989) 2255-2268.

[Fr 641 P. Freyd, Abelian Categories, (Harper & Row, New York, 1964)

[Go 861 J. S. Golan, Torsion Theories (Pitman Monographs and Survys in Pure and Applied Matematics, Longmaan Scientific & Technical, Harlow, 1986).

[HeSt 791 H. Herrlich and G.E. Strecker, Category Theory 2' Edition, (Heldermann-Verlag, Berlin, 1979).

[HeSaSt 871 H. Herrlich, G. Salicrup and G.E. Strecker, Factorizations, denseness, separation and relatively compact objects, Topology Appl. 2:' (1987) 157-169.

[Is 661 J.R. Isbell, Epimorphisms and dominions, Proc. Confer. Categorical Algebra, La Jolla 1965, (Springer, Berlin, 1966) 232-246.

[Ka 541 I. Kaplansky, Infinite abelian groups, Ann Arbor, 1954? Revised Edition 1969.

[La 711 J. Lambek, Torsion theories, additive semantics and rings of quotients,

Lecture Notes in Math 177, (Springer-Verlag, Berlin 1971)

[Ma 741 E.G. Manes, Compact Hausdorff objects, Gen. Topology Appl. 4 (1974) 341 -360.

[Or 791 A Orsatti, Introduzione ai gruppi abeliani astratti e topologici, Quademi dell'U.M.1. 8, (Pitagora Ed., Bologna, 1979).

[Pi 721 R. Pierce, Closure spaces with application to ring theory, Lecture :Votes in Maths Vol. 246 (Springer, Berlin, 1972), 516-616.

[Ro 681 J. Rotman, A completion functor for modules and algebras, J. of 4lgebra 9 (1968) 369-387.

[Th 831 W. Tholen, Factorizations, localizations, and the orthogonal sukiitegory problem, Math. Nachr. 114 (1983) 63-85.

Received: J u n c 1989

Revised: June 1990

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:08

16

Apr

il 20

13

Documents

Factorizations, injectivity and compactness in categories of modules