Self-projective modules over valuation rings

Arch. Math., Vol. 43, 332-339 (1984) 0003-889 X/84/4304-0332 $ 3.10/0 �9 1984 Birkh~iuser Verlag, Basel

Self-projective modules over valuation rings

By

PETER HERRMANN

I. Introduction. Quasiprojective or self-projective modules have been introduced by Miyashita [8] as a generalization of projective modules. Self-projective modules over various types of rings have been discussed by a number of authors ([2], [3], [4], [5]). In this paper, we shall investigate self-projective modules over valuation domains; i.e. domains whose ideals form a chain.

We characterize the valuation domains all of whose ideals are self-projective as the almost maximal valuation domains R; i.e. R/L is linearly compact for every ideal 0 ~ L of R. Generalizing a result by Rangaswamy and Vanaja [9], we show that the quotient field of R is a self-projective R-module if and only if R is complete in its R-topology. This leads to a characterization of maximal valuation domains; i.e. linearly compact valuation domains. Moreover, we give the structure of self-projective torsion-free modules of finite rank over an almost maximal valuation domain.

We were unable to get a full characterization of self-projective modules over valuation domains in general. However, we have obtained several results on the self-projectivity of direct sums of uniserial modules over valuation domains (see Theorems 4.7 and 5.4).

II. Preliminaries. All rings considered in this paper are commutat ive with 1 and all R-modules M, N, A, B, C . . . . are unital. We shall write N ~ M (N < M) to indicate that N is a (proper) submodule of M. A module M is said to be projective relative to B or B-projective if for every homomorph i sm f: M - ~ B/A there exists a homomorph i sm �9 : M -~ B such that f = can ~, where can: B -~ B/A is the canonical map, or equivalently, if the natural map can*: HomR(M, B) -~ HomR(M, B/A) is epic for every submodule A of B. A module M is called self-projective or quasiprojective if M is M-projective.

We state the following lemmas proved in [1]:

Lemma 2.1. (a) Let 0 ~ A ~ B ~ C ~ 0 be an exact sequence of R-modules. I f M is B-projective, then M is projective relative to A and C.

(b) I f a module M is projective relative to M i for each i in a finite index set I, then M is G Mi-projective. Moreover, if M is finitely generated, this holds for an arbitrary index

i e I

set I.

Lemma 2.2. Let {Mi: i E I} be a family of R-modules and B an R-module. Then (~ M i is B-projective if and only if each M i is B-projective. i~x

Vol. 43, 1 9 8 4 Self-projective modules over valuation nngs 333

III. Self-projeetivity of ideals. A commutative ring R is called a valuation ring if its ideals form a chain. We consider only valuation rings which are not fields. The set of zero-divisors in a valuation ring is a prime ideal, denoted by Z. Let Q be the ring of quotients of with respect to R - Z. For a proper ideal I of R, I*: = {r e R: r I < I} is a prime ideal containing I.

Lemma 3.1. Let I be an ideal of a valuation ring R containing Z properly. Then I is self-projective if and only if I is Q-projective.

P r o o f. "=>": Suppose we are given a homomorphism f: I -~ Q/U, U < Q. For an element r ~ I* - Z we have r . Q/U = Q/U and rI < I, hence r . I m ( f ) < I ra( f ) . Thus, f cannot be onto and we can write I r a ( f ) = V/U, where V is a proper submodule of Q, and therefore isomorphic to an ideal K of R. But Z < I implies that I contains a copy of R and hence an isomorphic copy of K. From Lemma 2.1 we conclude that I is V-projective and that we can, therefore, complete the following diagram

, I r

s

V ~a~, I m ( f )

Q ~ , Q / U

such that f = can �9 4. " ~ " : by Lemma 2.1. []

Lemma 3.2. Let I be an ideal o f a valuation domain R. Then the following statements are equivalent:

(a) I is self-projective.

(b) Ext~ (I, K) = 0 for every ideal K of R.

(c) For every ideal K of R, Q/K has the injective property relative to the sequence O ~ I -~ R ran ~ R / i _, O.

P r o o f. (a) ~:~ (b): By Lemma 3.1, (a) is equivalent to I being Q-projective. From this and the homology sequence

0 ~ HOmg (I, K) ~ U o m R (I, Q) ~ Horn R (I, Q/K)

Ext~R (I, K) ~ EXtlR (I, Q) = 0

the equivalence of (a) and (b) follows. (b) ~ (c): From the homology sequences

0 = Ext~ (R, K) ~ Ext~R (I, K) ~ Ext 2 (R/I, K) ~ Ext 2 (R, K) = 0

334 P. HERRMANN ARCH. MATH.

and 0 = Ext~ (R/I, Q) ~ ExtlR (R/I, Q/K) -~ Ext 2 (R/I, K) ~ Ext 2 (R/I, Q) = 0

we derive that Ext~ (I, K) ~ Ext~ (R/I, Q/K). But the latter vanishes for all K < R if and only if (c) holds. []

Now we can prove the following theorem which characterizes valuation domains whose ideals are self-projective. Recall that a valuation ring R is called almost maximal if R/L is linearly compact for every ideal L 4= 0 of R, and maximal if R is linearly compact in the discrete topology.

Theorem 3.3. For a valuation domain R the following statements are equivalent:

(a) R is almost maximal. (b) Every ideal of R is self-projective.

P r o o f. R is almost maximal if and only if Q/K is injective for every ideal K of R, by Matlis [7]. Using Lemma 3.2 and Baer's criterion we see that this is equivalent to (b). []

We can consider a commutat ive domain R as a topological ring equipped with its R-topology; i.e. the family {rR: 0 4= r E R} forms a subbase for the open neighborhoods of zero. Matlis proved (Theorem 14 in [6]) that a domain R is complete in its R-topology if and only if Ext~ (Q, I) = 0 for every ideal I of R.

This leads to the following result (cp Lemma 5.1 in [9] and its generalization to non-commutat ive Dedekind prime rings by S. Singh [10]):

Theorem 3.4. For a valuation domain R, the following statements are equivalent:

(a) R is complete in the R-topology. (b) Q is self-projective.

P r o o f. Let K be a proper submodule of Q. F rom the exact sequence

0 ~ Horn R (Q, K) ~ Horn R (Q) --* n o m R (Q, Q/K)

EXt~R (Q, K) ~ EXt~R (Q, Q) = 0

and the fact that every proper submodule of Q is isomorphic to an ideal of R, it follows that Q is self-projective if and only if Ext~ (Q, 1) = 0 for every ideal I of R. Thus, Matlis ' theorem mentioned above finishes the proof.

It is well-known that maximal valuation valuation domains which are complete in the theorems, we obtain:

[]

domains are exactly the almost maximal R-topology. Thus, combining the last two

Theorem 3.5. Let R be a valuation domain. Then R is maximal if and only if every submodule of Q is self-projective.

IV. Self-projective Torsion-free Modules. A module M over a domain R is called torsion-free if 0 4= m ~ M and 0 4= r ~ R imply mr 4= O. The rank of a torsion-free module M, denoted by rk (M), is the cardinality of a maximal R-independent set contained in M.

Vol. 43, 1 9 8 4 Self-projective modules over valuation rings 335

We state the following corollary to Lemma 3.2 for easy reference:

Corollary 4.1. I f I is a self-projective ideal of a valuation domain R and M is a torsion-free module of finite rank, then Ext~ (I, M) -- 0.

P r o o f. By induction on the rank of M and using Lemma 3.2. []

We list a few well-known results about self-projective modules.

Lemma 4.2. Let R be an arbitrary local ring and M a self-projective R-module possessing a projective cover. Then, M is isomorphic to a direct sum of copies of R/I, for some fixed two-sided ideal I of R.

P r o o f. See A. Koehler [5]. []

As finitely generated modules over local rings possess projective covers, we derive that a finitely generated self-projective module over a valuation ring R is a finite direct sum of copies of R/I, for some ideal I of R.

For an infinite cardinal N~, a module M is said to be N~-generated, if M can be generated by N~ elements but not by a subset of M with less than N~ elements.

Lemma 4.3. Let R be a domain and M a self-projective torsion-free R-module which is N~-generated. Then

(a) rk (M) >= card (R) implies that M is projective. (b) rk(M) = N~ implies that M is projective.

P r o o f. (a) is proved in [9]. (b) Observe that M is an epimorphic image of its free submodule, generated by a

maximal R-independent set and use Lemma 4.4 in [9]. []

Thus, over valuation domains R, "big" selfprojective torsion-free modules are projective, and hence free.

We turn our attention to torsion-free modules of finite rank. Our results are concerned with almost maximal valuation domains. Recall that a torsion-free module M over a domain R is called divisible if, for every 0 ~: r ~ R, M r = M, and reduced if 0 is the only divisible submodule of M. A torsion-free module M which is a direct sum of rank 1 submodules, is called completely decomposable.

Lemma 4.4. Let M be a torsion-free module of finite rank over a valuation domain R.

(a) I f R is almost maximal, but not maximal, then M is self-projective if and only if M is reduced completely decomposable.

(b) I f R is maximal, then M is self-projective.

P r o o f. (a) " ~ ' : Direct summands of self-projectives are again self-projective. There- fore, as R is not maximal, M has to be reduced by Theorem 3.4; in particular, there exists an 0 :~ r E R with r M ~e M. Let B be a basic submodule of the finite rank module M, i.e. B is a direct sum of rank 1 submodules B 1 . . . . , B, and M / B is divisible. Thus, especially


for the 0 + r e R selected above, it holds M = B~ + .. . + B, + r M and it follows readily that M = A + C , where A is of r a n k l and C is a proper submodule of M. By H. Z6schinger [11], there exists an endomorphism f of M with Im(f)__< A and Im(idM - - f ) <_- C. f cannot be 0, as C is proper; thus, 0 4 Im (f) is torsion-free of rank 1 and not divisible, as M is reduced, and therefore isomorphic to an ideal of R. Theorem 3.3 and Corollary 4.1 imply ExtlR (Im (f), Ker (f)) = 0. We conclude that M ~ , M / K e r ( f )

Im (f) is splitting, i.e. M ~ Im (f) @ Ker (f). Induction now finishes the proof. " ~ ' : If M = M 1 O)... O) M, with M~ ideals of R, then using Theorem 3.3 and Lemma

3.1, we derive that M~ is Mfprojective for all 1 =< i , j <= n. Lemmas 2.1 and 2.2 imply that M is self-projective.

(b) Matlis proved (Theorem 51 in [6]) that finite rank torsion-free modules over maximal valuation domains are completely decomposable. Therefore, the same proof as in (a) " ~ " applies, with obvious modifications. []

Lemma 4.3 shows that self-projective, but not projective modules over a domain have smaller rank than the number of its generators or the cardinality of R; in particular, arbitrary direct sums of non-cyclic ideals are not self-projective. Our next goal is to characterize self-projective completely decomposable modules over valuation domains. Recall that a module is called uniserial if its submodules form a chain.

We start with the interesting lemma about uniserial modules:

Lemma 4.5. Let U be an N~-generated uniserial module over an arbitrary ring R, {Mi: i ~ I} a family o f R-modules with card (I) < N~ and N a submodule o f M = O) M i. Then,

for every homomorphism f : U ~ M / N there exists a finite subset K o f I such that Im (f) _<_ can (O) Mi), where can: M -~ M / N is the canonical map.

i~K

P r o o f. It is an elementary fact that an epimorphic image of an N~-generated uniserial module is again N~-generated, hence, Im (f) is N~-generated. Let K be a subset of I of smallest cardinality such that Im (f) __< can (O) M~). Assume, by way of contradiction,

iEK

that K is infinite, card (K) = N~, ~r > 0. Let co~ be the initial ordinal of cardinality N~; without loss of generality K = co, can be assumed. Each 0 < co~ has cardinality less than N,, therefore, Im (f) is not contained in can (O) M~). So, for each 0 < c% we can pick

~tEQ

an element x Q ~ I m ( f ) - - c a n ( 0 ) M ~ ) . Since I m ( f ) is uniserial, it follows that

I m ( f ) c~can(o)M~)<xQR. As U O) M ~ = @ M~, we obtain

Im (f) _-< ( U can (O) M~) c~ Im (f)) _-< ~ x e R < Im (f).

Thus, Im ( f ) = U xQR, a contradiction to the fact that Im (f) is N~-generated with p < toe,

N~ > card(I) > N~ = card (c%). []

Corollary 4.6. Let U be an N~-generated uniserial module over an arbitrary ring R, which is projective relative to modules Mi, i e I, I an index set o f cardinality less than N~. Then U is projective relative to M : = O) M i .

i~I


P r o o f . Suppose we are given a homomorphism f : U--+M/N, N < M. By Lemma 4.5, there exists a finite subset K of I such that Im (f) < can ( | Mi), where

i e K

can: M --, M / N is the canonical map. Lemma 2.1 implies that there exists a ~b: U --* @ M~ such that f = can �9 ~ . [] i~:

We can now prove the following:

Theorem 4.7. Let M = O)Mi be a completely decomposable module over a valuation domain R. ieI

Then the following statements are equivalent:

(a) M is self-projective. (b) (1) Each M s is self-projective.

(2) I f M i is N~-generated, where N~, is an infinite cardinal, then N~, > card (I).

P r o o f. (a) ~ (b): (1) follows by Lemmas 2.1 and 2.2. To prove (2) suppose for a contradiction that there exists an Mi which is N~-generated, c~i __> 0, such that N,, < card (I) = rk(M). Let F < M be a free submodule of rank N~. Then, M i is isomorphic to an epimorphic image of F and M s is F-projective, by Lemmas 2.1 and 2.2. Hence, M~ is free uniserial, a contradiction to M~ not being finitely generated.

(b) ~ (a): It suffices to show that each M i is M-projective, by Lemma 2.2. (1) and Lemma 3.1 imply that each Mi is Q-projective; hence, each M i is MFprojective for every M j < Q. If M~ is finitely generated, then M i is projective and thus, trivially projective relative to M; if M~ is N~-generated for an infinite cardinal N~, then Corollary 4.6 applies. []

V. Selfprojeetivity of direct sums of uniserial modules. We shall need the rather technical

Lemma 5.1. Let R be an arbitrary ring, N < M, B R-modules and M B-projective. Then,

the following statements are equivalent:

(a) M / N is B-projective. (b) For every homomorphism f : M --* B there exists a homomorphism ~: M / N ~ B such

that M = N + Ker ( f - �9 �9 can~), where canM: M ~ M / N is the canonical map.

P r o of . (a) ~ (b): For a given homomorphism f : M - ~ B consider the solid part of the diagram with commutative square

M . . . . ) M / N

B ~ B/f(N),

wheref(n~) : = f (m) and canM, cane are the canonical maps. By hypothesis, there exists an �9 : M / N ~ B such that the lower triangle commutes. F rom c a n ~ ( f - ~ , canM)= can B . f - - f - can M ---- 0, it follows that I m ( f - ~ . canM) <=f(N). Hence, for every m e M there exists an n e N such that ( f - ~ . c a n M ) ( m ) = f ( n ) = ( f - - ~ . c a n M ) ( n ) ; i.e. m -- n ~ Ker ( f - - ~ . canM). Thus, m = (m - n) + n s Ker ( f - - ~ . canM) + N.

Archlv der Mathematlk 43 22


(b) ~ (a): Suppose the solid par t of the d iagram

M . . . . , M / N

i s s l (o ' @ s f

~,te J

B ~ B/C

is given. M B-projective implies that there exists a homomorph i sm q~: M --* B such that can e �9 ~o = f . can M. By hypothesis, there exists a homomorph i sm ~: M / N ~ B such that M = Ker (~o - ~ �9 canM) + N. It follows readily that f . canM = canB. 4~ �9 can M and as can M is right cancellable, f = can e �9 4~. []

A submodule N of M is called small in M, denoted by N < ~ M, if M = A + N, A < M, implies A = M. Observe that every proper submodule of a uniserial module M is small in M.

Corollary 5.2. Let N i <= ~ M 1 and N E <= M 2 be modules over an arbitrary ring R. I f M i is ME-projective , then the following statements are equivalent:

(a) M 1 / N 1 is ME/N2-projective. (b) Every homomorphism of M 1 into M 2 carries N 1 into N 2.

P r o o f. (a) ~ (b): M i is ME/NE-projective , by Lemma 2.1. Thus, the previous Lemma 5.1 for B = M E / N 2 implies: for every homomorph i sm f : M i ~ M E , there exists a homomorphism ~: M i / N 1 ~ ME/N 2 such that M 1 = ker(can 2 - f - ~ �9 cana) + Na, where cant: Mi ~ M~/Ni, i = 1, 2, are the canonical maps. F r o m Ni _-<~ M1, it follows that can2 . f = ~ . cani . Hence, can 2 .f(N1) = 0 and therefore f (Na) _-< NE.

(b) =~ (a): If we have the solid par t of the d iagram

~, l . . . . , M 1 / N i

le

M2 . . . . , ME/N 2 a ' C,

then there exists a map @: M1 ~ M 2 such that the rectangle commutes, as M 1 is ME-projective. By hypothesis, �9 induces a homomorph i sm cp: M 1 / N i ~ M E / N 2 such that f = g . ~o. []

F o r two submodules J, K of the quotient field Q of a valuat ion domain R, we define (J : K): = {q e Q: q K <= J}. It is easy to see that Horn R (K, I ) = (J: K). A uniserial module of the form J /K, K _< J _< Q, over an valuat ion domain R is called standard uniserial. It is an easy exercise to show that, over an almost maximal valuat ion domain, every uniserial module is s tandard uniserial.

Lemma 5.3. Let {Ui = Ji/Ki 4: 0, i e I} be a family o f standard uniserial modules over a valuation domain R and let �9 Ji be self-projective. The following statements are equivalent:

i E l

(a) U = @ U i is self-projective. i ~ I

(b) (Ji : Ji) <= (Ki: Ki) for all i, j e I .


P r o o f. (a) ~ (b): Once more using Lemmas 2.1 and 2.2, it follows that J J K i is J j K T p r o j e c t i v e , for all i , j ~ I. From Corollary 5.2, (b) follows at once.

(b) ~ (a): By Lemma 2.2, we only need to show that Ui is U-projective for every i 6 L Corollary 5.2 implies that Ui is Ufprojective, for all i , j ~ I. If U~ is finitely generated, then Lemma 2.1 yields the assertion. If U~ is N~ -generated, N~, an infinite cardinal, then - as 0) Ji is self-projective - Theorem 4.7 implies that card (I) < N~. Thus, Corollary 4.6 t e l

applies. []

As finite direct sums of (proper) submodules of Q are self-projective over (almost) maximal valuation domains (Lemma 4.4), we obtain:

Theorem 5.4. Le t R be an (almost) maximal valuation domain and U i = Ji/Ki +-0, n

i = 1 . . . . . n, a f ini te number o f uniserial modules with J~ < Q (Ji < Q). Then, �9 Ui is self-projective i f and only i f (Ji:Jj) < (Ki:K2) for all 1 < i , j < n. i=1

References

[1] F. W. ANDERSON and K. R. FULLER, Rings and Categories of Modules. New York-Heidelberg- Berlin 1973.

[2] L. FUCHS and K. M. RaNGASWAMV, Quasiprojective abelian groups. Bull. Soc. Math. France, 98, 5-8 (1970).

[3] J. S. GOLAN, Characterisation of rings using quasiprojective modules II. Proc. Amer. Math. Soc. 28, 337-343 (1971); III, 31,401-408 (1972).

[4] D. A. HILt, Artinian rings in which one sided ideals are quasi-projective. Acta Math. Acad. Sci. Hungar. 40, 11-20 (1982).

[5] A. KOEHLER, Quasi-projective and quasi-injective modules. Pacific J. Math. 36, 713-720 (1971).

[6] E. MATLIS, Torsion-free modules. Chicago 1972. [7] E. MATLIS, Injective modules over Prfifer rings. Nagoya Math. 15, 57-69 (1959). [8] Y. MIYASmXn, On quasi-projective modules, perfect modules and a theorem for modular

lattices. J. Fac. Sci. Hokkaido Univ. 19, 86-110 (1966). [9] K. M. RANGASWAMY and N. VANAJA, Quasi-projectives in abelian and module categories.

Pacific J. Math. 43, 221-238 (1972). [10] S. SINGH, Quasi-projectiveandquasi-injectivemodules over hereditary Noetherian prime rings.

Canad. J. Math. 26, 1173-1185 (1974). [11] H. ZOSCHINGER, Komplemente als direkte Summanden. Arch. Math. 25, 241-253 (1974).

Anschrift des Autors:

Peter Herrmann Mathematisches Institut der Universit/it Miinchen Theresienstr. 39 D-8000 Miinchen 2

Eingegangen am 14. 11. 1983 *)

*) Eine Neufassung ging am 3.4. 1984 ein.

22*

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Self-projective modules over valuation rings