Results in MathematicsVol. 20 (1991)

0378-6218/91/020530-08$1.50+0.20/0(c) 1991 Birkhauser Verlag, Basel

RINGS OF STABLE RANK 2 ARE BARBILIAN RINGS.

Werner LeiBner

Dedicated to my highly esteemed teacher WALTER BENZ

on occasion of his 60th birthday

Summary. Let M R a free un itary modul. A non-empty subset B of M R is bydefinition a Barblllan set iffB'" : = {F c B I F is a free generating system of M Rand #F > I} satisfies:(Bt) Each u t E B may be completed to {up uz""} E B'"(Bz) {u t ' uz' '' '} E B'" implies {u t + uzO: , uz' .. .} E B'" for all 0: E R.For each Barbilian set B c M R holds : B c B rn a x : = {u E M Iu may be com­pleted to a free generating system F of MR and #F > 1} and Brn a x is aBarbilian set i ff Brna x i Il.A ring R is by defin ition a BarbllJan ring iff each Barbilian set B over anyfree unitary R-module M R coincides with Brn a x ' We show that all rings ofstable rank 2 are Barbilian rings. The class of rings of stable rank 2 coversfor instance the class of all s e m ip r im a ry r ings which contains the classo f all finite rings with identity . This answers two questions of W . BENZ,posed a few years ago to the author .

1. Introduction. Let VF a right vectors pace of arbitrary dimension d > 1

and B c VF a Barbilian set as defined above. We call the elements of

VF = P Polnt.s and those of L :={L~ :=a + uFla E VF ' U E B} LInes. A point

p is s a id to be non-nelghboured to q - in signs p $ q - iff p - q E Band

a line L~ is by definition parallel to Lb - in signs L~IILb - iff uF = vF. The

quadruplet (P, L, $, II) constitutes an affine geometry over VF relating to

B, shortly denoted as srrcvF' B). Of course, the conditions (Bt) and (Bz)

of an Barbilian set im p ly - in the vectorspace case - B = Brn a x = VF - {O}.

Leissner 531

Therefore, a ell b iff a :j: b. L = {G~la. u E VF with b :j: o} and Af[(Vpo B)

= (P. L. :j:. II) is nothing else but an unusual representation of the affine

geometry over the vectors pace V po

Replacing the field F by a ring R with identity and VF by a free unitary

right R-modul M R' one gets an algebraically defined class of affine

geometries A{f(MR' B), which - up to isomorphism - coincides with the

class of all affine Barbilian spaces (LEISSNER. SEVERIN. WOLF [4]) .

In this more general situation it may happen that there exist Barbilian

sets Be M R , s. t. B :j: Brna x ' For instance. a result of COHN [2] implies

that the ring R of integers of Q(F19) is a principal ideal domain, but that

nevertheless R2 contains Barbilian sets B ~ Brna x

(c. f. LEISSNER [3]>. On

the other hand . for several classes of rings - for in s t a n ce Euclidean rings ­

this cannot happen, as was proved by BENZ Ct].

In the following, all rings under consideration are supposed to have an

identity and their group of two-sided units will be denoted by U.

2. Let R a ring with identity 1. For arbitrarily given

0 . j, n , ex) E IN x IN x IN x R, such that i :j: j and l , j s: n ,

the matrices (a,,(..L) E Rn xn satisfying

j1. if v = [!

a V(..L 0: . if (v .u) = H.i>O. otherwise

are ca ll ed elementary matrices and will be denoted by E~j)(o:).

The functional equation

- applied to ~ : = -0: - shows that the set En(R) of all elementary n x n

- m a t r ice s is contained in GLn(R) and that the set of all finite products

of elements of En(R) constitutes a subgroup of GLn(R), denoted a s GEn(R) .

An elementary calculation shows that for all 0: 2 , .. .• O:n E R

532 Leissner

l1

oc 2 • . "n]1 n(2.1) = II E(n)(oc ) E GEn(R)

\1= 2 1\1 \I

and

l~2

. I ]

n(2.2) II E(n )(oc ) E GEn (R)

\1=2 \11 \I

oc n

Now , let M R a f ree , unitary , r ig h t Re-modu l , B C M R a Ba rbilian s e t a s de­

fined in the summary and Brna x : = {u E M R I u is member of some basis F

of M R with IIF > t} .

Then .

(2.3) B c Brna x

and

(2.4) Bm a x

is a Bar b il ian set iff Bm a x :j: II.

Let n different elements up . . .• un of some (Ui)iEI E B'" and E~~)(oc) E E n(R)a rbit rarily g iven. Property (B 2 ) of a Barbil ian set shows that

(u 1• . .. • un) ' E~~) («) = (u 1• . . . • ur

+ ukoc •.. . , un) are n d ifferent elements

of some ( Vi)iEI E B'" w ith

Repeating this step finite many t imes one gets

(2.5) If u 1• . .. , un are n different el ement s o f some FE B'" and

D E GEn( R) t hen (u , • . ..• un ) D = : (w , . .. . . w n ) are agai n n different

e lements of so me F' E B"' .

3. Let R a ring w ith id e n ti t y 1.

(3.1) (91' . .. • 9n ) E Rn is ca lle d un imodular i f an d only i f there exist

A 1, . . .• An E R S .t . A191 + . . . + An 9n = 1.

Leissner 533

(3.2) R is said to ha ve stable rank 2 if and only if for each unimodular

(9 1, (2) E R2 there e xist 0:. ~ E R s.t . 0:(91 + ~(2) = 1.

Let R a ring with stable rank 2 and suppose ~. Tl E R satisfy ~Tl = 1. Then

(~. 1 - Tl~) is unimodular and there exist 0:. ~ E R S.t. o:(~ + ~(t - Tl~» = 1.

Now. ~ + ~(t - TlO =: C is a two-sided unit of R because O:C = 1 = CTl. It

follows that 0: = Tl. Therefore. Tl is a two sided unit . too. and we get a

result of Kaplanski and Lenstra .

(3.3) Each ring of stable rank 2 is a 1-finite ring. i .e .

CTl = 1 impl ies TlC = 1.

With respect to (3 .3) the definition (3 .2) is equivalent to

(3.4) R is sa id to ha ve stable rank 2 if and only i f for ea ch unimodular

(C1• C2 ) E R2 there ex ists some ~2 E R. s .t. 91 + ~292 is a two s ided

un it in R.

For the remainder of section 3 let R always denote a ring of stable rank 2.

(3.5) Let n E IN. For each unimodular (91 ' . . .• 9 n) E Rn there exist

~2 ' ... . ~n E R. s .t . 9 1 + ~292 + .. . + ~n9n E U.

Proof: (3 .5) holds because of (3 .t) and (3.4) for n = 1 and n = 2. Now,

suppose that (3.5) holds for a fixed n ;;, 2 and consider a unimodular

(91' ... , 9 n-1' 9 n• 9 n+1) E Rn +1. According to (3.0 there exist

A1•.. .• An- 1• An' An+ 1 E R, s .t. A191 + ••• + An- 19n-1 + An9n + An- 1 9 n+1 = 1.

Therefore , (91' ... , 9 n-1' An9n + An+ 19 n+1) E Rn ist unimodular, too .

By the induction assumption there exist ~2' ••• , ~n E R. such that

91 + ~292 + ... + ~n(An9n + An+ 19 n+1) E U, proving that (3 .5) holds forall n E IN.

Let (~1 ' •• • , ~n) E R" , n E IN, unimodular. Because of (3 .5) there exist

~2' . . . , ~n E R, S.t. ~1 + ~2~2 + . •. + ~n~n = : 9 E U, Le ,

534

Now .

Leissner

and

Le , (obey (2 .0 and (2.2» there exist A. B. C E GEn(Rl. such that

CBA' ( ~1' •..• ~nlT = (0 . .... O. OT and with (CBAl- 1 = :D we get

(3.6) For each unimodular ( ~ 1' . . . • ~n l E Rn there exists D E GEn(R l .

S. t. (~1' . . . • ~nl T = DW . . . .. 0, nT .

Finally . let B an arbitrary Barbilian set of M R and choose a fixed

(ull l EI E B"'. Let v1 an arbitrary element of Brna x ' Then. v1 = ~ UI~I withlEI

unique ~I E R and almost all ~I = O.

Let for convenience of notation {I E I I~I :l: O} - . {l , 2•... • n} , I.e , letn

v 1 = L UI ~I'1=1

Because of the definition of Brna x ' v 1 may be completed to a bas is (v jl j EJ

of M R with t E 1.Therefore. for each i E u.z .....n} holds

u l = ~ Vj~jl with unique ~jl E R and almost all ~jl = O.jEJ

Let {j E 1I ~j l :I: o} _. 11 and - for convenience of notation -

11 U 12 U .. . U 1n

U {t} _. {t. 2... .. m} ,

Leissner

It follows

n

v t = L Ul ~l =l = t

535

n

whence I ~t . ~ . = 1. Hence ( ~t ..... ~ ) is unimodular and (3.6) guaranteesl=t J J n

the existence of some 0 E GEn(R). s .t. (~t•... • ~n)T = 0<0 • ...• O. nT .

Let (u t • • ••• un)O = : (w t . .... w n). Then {Wt..... w n} C B because of (2 .5) .

and we get

(u,.. un' []J =(u ,•...• unlDmTherefore. B ::) Brna x ' and because of (2.3) B = B

rna x' which proves the

Theorem : Rings of stable rank 2 are Barbilian rings .

4. In this final section we quote from Veldkamp [6] some known results

co nc e r n in g the class of rings with stable rank 2 and include for conveniance

of the reader their short proofs .

Immediately . one gets

(4.1) The di rect prod uct o f an arbi t ra ry number o f r in g s w ith s tab le rank

2 ha s aga in s t abl e rank 2 .

(4.2) Ea ch homomorph ic image o f a r in g with stab le rank 2 has again

stable rank 2 .

Proof: Let

an epimorphism of a ring R onto a ring 5. Consider some unimodular

(9t . ( 2 ) E 52 . There are A t. A2 E R. s .t , AtP t + A2 P 2 = 1 + II with some

II E ked<pl. Therefore . (Pt . A2 P 2 - II) E R2 is unimodular . too .

536 Leissner

Now, in case R has stable rank 2 , there exist a ,~ E R, s.t. a(91 + ~ ()' 2 9 2 - (l»

= 1, whence a(Pl + ~"I2 (2) = 1, I.e , 5 has stable rank 2.

(4 .1) and (4 .2) imply

(4.3) T he direc t product of an arbitrary number of r ings Rex has stable

r ank 2 if and onl y if each ring Rex has stable rank 2 .

(4.4) Let R a ring and J(R) its Jacobson r adical. T hen R has sta b le rank 2

i f and on ly i f R/J (R) =: R has stab le r ank 2 .

Proof: If R has stable rank 2 one gets immediately from (4 .2) that R has

stable rank 2 as well . - Conversely, assume that R has stable rank 2 and

let

the canonical epimorphism from R onto R. Consider a unimodular

(91 ' (2) E R2 . There are A1, A 2 E R, s .t. A19 1 + A292 = 1, whence A19 1 + A 2 9 2 = 1.

Therefore (Pl ' (2) E R is unimodular , too . Because R has stable rank 2,

there must eXist~, if E R s .t. ~(Pl + [3(2) =1, whence a(91 + ~ (2) = 1 + (l with

some (l E J(R) or (t + (l)- l a ( 9 1 + ~(2) = 1, recall ing that J(R) = {(l E R I1 + A(l E U for all A E R}.

(4.5) The r ing Mn(D) o f all nxn-matr ices over an arbitrary (no t necessarily

commutat ive) fie ld D has stab le rank 2 .

Proof: Let the elements of Mn (0) act as linear transformations on the right

vectors pace onxl and consider an unimodular (A,B) E Mn ( 0 ) 2 . The relation

XA + YB = I implies that kerf A) n kert B) = O. Therefore on xl splits into

onxl = kerf A) @ L with L ::> ke rfB).

Now, there exists an invertible P E Mn(D) and some Q E Mn(O), such that

PA is the projection onto Land QB the projection onto ke rt A), whence

PA + QB = I , which implies that A + P- 1QB = p- 1 is invertible.

Finally let us recall the famous Artln-Wedderburn Theorem (c .F. McCOY

[5] , Theorem 5 .59 and Corollary 6.26):

Le t R a non zero r ing such that J(R) = 0 and every s t r ic t ly decre­

asing se q uence of right ideals o f R is f in ite . Then R is iso m or ph ic

to the d irect product of a f inite number o f r ings Rex' each o f which

is a complete matri x ring Mn(ex J(D ex) over some no t ne cessarily com ­

mutat ive fie ld Dex .

Leissner 537

Obeying (4.0-(4.5) and J (R~(R») = 0, we conclude from the Artin-Wed­

derburn Theorem

(4.6) Eac h semiprimary ring R (j .e . each ring . suc h t hat every s t ri ctl y

de c r eas ing sequence of right idea ls of R/J(R ) is f inite ) has s tab le

rank 2 .

As a corollary of (4.6) one gets

(4.7) Ea ch finite ri ng R has stabl e r ank 2 .

REFERENCES

[1] BENZ, W.: On Barbilian Domains over Commutati ve Rings.

J . Geometry 12(979), 146-151

[2] COHN, P.M.: On the Structure of the GL z of a Ring .

Publ. Math . I.H .E .S. 30(966) , 5-53

[3] LEISSNER, W. : Barbilianbere iche ,

219-224 in H. J . Arnold et al. (eds.): Be itrage z ur Geometris chen

Algebra. Basel 197 4

[4] LEISSNER, W . , SEVERIN, R., WOLF , K.: Affine Geometry o ver Fr e e

Unitary Modules. J. Geometry 25(985), 101-120

[5] McCOY, N.H.: The Theory of Rings.

New York 1964

[6] VELDKAMP, F.D.: Projective Ring Planes and Their Homomorphisms,

289-350 in R. Kaya et al. (eds .) : Rings and Geometry. Dor,.drecht/

Boston 1985

Einge~angen am 5. Mai 1991

Werner LeiBner

Fachbereich 6

der Il ntver-stta t Oldenburg

Arnme r l a nde r Heerstr . 114-118

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