ON SEMIREGULAR RINGS

NEW ZEALAND JOURNAL OF MATHEMATICS Volume 32 (2003), 11-20

ON SEMIREGULAR RINGS

H u a n y i n C h e n a n d M i a o s e n C h e n

(Received January 2002)

Abstract. A ring R is called semiregular if R /J (R ) is regular and idempotents lift modulo J(i?). Let { e i , • • • , en } be a complete orthogonal set of idempotents of R. It is shown that R is semiregular if and only if all eiR ej are semiregular. Also we extend this result to unit semiregular rings and semiregular rings satisfying unit 1-stable range by different routes.

1. Introduction

A ring R is said to be regular provided that for any x £ R there exists a y £ R such that x = xyx. A ring R is said to be semiregular if R/J(R) is regular and idempotents lift modulo J(R), where J(R) denotes the Jacobson radical of R (see [8]). The class of semiregular rings is very large. For example, every regular ring and every right quasi-injective ring are semiregular. Let e, / G R be idempotents. Following W.K. Nicholson [8], we call eR f semiregular if for any x € e R f , there exists a y e fR e such that x — xyx € J(i?) and y = yxy. Let {e i,-- - ,en} be a complete orthogonal set of idempotents of R. In this paper, we show that R is semiregular if and only if there exists a complete orthogonal set {e i ,- - - ,en} of idempotents such that all eiRej semiregular. As applications, we investigate semiregularity of Morita contexts and trivial extensions.

A ring R is said to have stable range one provided that aR + bR = R implies a + by € U(i?) for a y £ R. R is said to be unit semiregular if R is a semiregular ring having stable range one. If the element y G R is invertible, R is said to satisfy unit 1-stable range. We will investigate stable range one and unit 1-stable range for semiregularity. It is shown that R is unit semiregular if and only if there exists a complete orthogonal set {ei,-- - ,en} of idempotents such that all eiRej are unit semiregular. By a different route, we also generalize this result to unit 1-stable range for semiregularity. That is, we prove that R is a semiregular ring satisfing unit 1-stable range if and only if there exists a complete orthogonal set {ei, • • • , en} of idempotents such that all eiRej are semiregular rings satisfying unit 1-stable range.

Throughout, all rings are associative with identity and all modules are unitary. We use 3(R) to denote the Jacobson radical of R. Let M and N be right i?-modules. The notation M <® N means that M is isomorphic to a direct summand of N.

2000 A M S Mathematics Subject Classification: 16E50, 16U99.

12 HUANYIN CHEN AND MIAOSEN CHEN

2. Semiregularity

Lemma 2.1. Let {e i ,- - - ,en} 6e a complete orthogonal set of idempotents. If all eiRei are semiregular rings, then idempotents lift modulo any ideal of R.

Proof. Since e\Re\ is a semiregular ring, e\Re\/ Z{e\Rei) is regular and idempotents lift modulo J(eiitei). So eiRei/J(eiRei) is an exchange ring. It follows by [1, Corollary 2.3] that e\Rei is also an exchange ring. Likewise, e^Re2 ̂• • • , enRen are all exchange rings. Hence R = e\R © • • • © enR is an exchange ring. Thus the lemma is true by using [1, Corollary 2.3] again. □

A finite orthogonal set of idempotents e\, in case e\ H------- 1- en — 1 G R.

, en in a ring R is said to be complete

Theorem 2.2. Let {e i, • • • ,en} be a complete orthogonal set of idempotents of R. Then the following are equivalent:(1) R is semiregular.

(2) All eiRej are semiregular.

Proof. (1) => (2) Given any x G eiRej, by [8, Proposition 2.2], we have a y € R such that x — xyx G J(jR) and y = yxy. Hence x — x(ejyei)x = x — xyx = e i(x—xyx)ej G eiJ(R)ej and ejyei = ejyxyei — (ejyei)x(ejyei). By [8, Proposition 2.2] again, we show that eiRej is semiregular, as required.

(2) =4- (1) Since {e i, • • • , en} is a complete orthogonal set of idempotents and all eiRej are semiregular, {eT, • • • ,e^} is a complete orthogonal set of idempotents of R/3(R). In view of [8, Proposition 2.2], one easily checks that all eiRej/eiJ(R)ej = ei(R/J(R))ej are regular. It follows by [5, Lemma 1.6] that R/J(R) is regular. In addition, idempotents lift modulo J(i?) from Lemma 2.1. Therefore we conclude that R is semiregular. □

Let ei, e2, • • • , en G R be idempotents. One easily checks that

(e\Re\ ••• eiR en\

\enRe i' /e ir n e i Cl r ln^n^

rij G R{ 1 < i , j < n)k \6l^nl6l Clrnn -̂nJ

forms a ring with the identity diag(ei, • • • , en). As an application of Theorem 2.2, we now derive the following.

Corollary 2.3. Let ei, • • • , en be idempotents of a ring R. Then the following are equivalent:(1) All eiRej are semiregular.

(e\Rei ■■■ e iR e n \

: •. ; ) is semiregular.

enR e 1 • •• enR en /

ON SEMIREGULAR RINGS 13

e\Re i ei Rer

Proof. Set T = | ; •. ;\ ••• CjiRsn /

idempotents g\, /12 £ T such that/0 exRe^

0 0

Suppose that T is semiregular. Then we have

0

\o 0 0

o\0

0

0

— 9 iTh2.

Clearly, <?iX7i2 is also semiregular. For any x G eiR e2-, we have some y E e^Rei such that

/ ° X o ••• o\0 0 0 ••• 0

0 0

\ 0 0 0 ••• V

/ 0 x 0 ° \0 0 0 0

0 : : 0

^ 0 0 0 0 /

and

( o 0 0 ••• 0\

y 0 0 ••• 0

0 00 0 ••• 0 /

/ 0 0 0

y 0 0

0

\ o 0 0

/ o 0 0y 0 0

0 : :\0 0 0

0 \ ( 0 X 0 • • 0\0 0 0 0 • • 0

0 0 • 0

Vo 0 0 • • V

G gi3{T)h,2

• ° \ • 0

. 0

• 0/It is easy to verify that

J(T) = J(diag(ei,

0 : VO 0 0

0\0

0

0/

/ o 0 0y 0 0

0 : :\0 0 0

0\0

0

0/

,en)Mn(JR)diag(ei,-- - ,c n)) = diag(ei, • • • ,en)J(M n(jR))diag(ei, • • • ,en) = diag(ei, • • • , e„)M n(J(i?))diag(ei, • • • ,e„)

eiJ(i2)ei eiJ (R)e2 ei3(R)en\

,en3(R)ei cnJ(R')c2 cnJ(̂ R̂ cnJ

Thus x — xyx G e\J(R)e2 and y = yxy, and then e\Re2 is semiregular. Similarly, we show that all eiRej are semiregular.

Conversely, choose f i = diag(ei, 0, • • • , 0), • • •, f n = diag(0, • • • ,0, en). Then { / i , • • • , / „ } is a complete orthogonal set of idempotents of T. Consider / i T / 2.

Clearly, f 1T f2 =0 eiRe2 0

.0 0 0


( 0 x 0 ••• 0 \: : : : I G f\Tf2, we have a y e e2Re\ such that y — yxy and 0 0 0 - 0 /

x — xyx € eiJ(i?)e2 because e\Re2 is semiregular. Analogously to the consideration above, we verify that

( 0 X 0 °\

\° 0 0 V

/o X 0

\0 0 0

and that

/ ° 0 0 0\y 0 0 0

V> 0 0 0/

(Q 0 0y 0 0

°̂ \ /O * 0

Vo 0 0 0 /\0 0 0

0 N

! I e / i J ( T ) /2,

0 ,

/0 0 0 y 0 0

\0 0 0 •

Therefore / 1T /2 is semiregular2.2, we get the result.

Vo 0 0

(0 0 0 • 'y 0 0 • • 0

0 0 • • 0/Likewise, all fiT fj are semiregular. Using Theorem

□• © enR),We note that the ring in (2) is in fact isomorphic to End#(eii?

where e\R ® • • • © enR denotes the external direct sum.

Corollary 2.4. Let ei,ring

is semiregular.

, en be idempotents of a semiregular ring R. Then the

( e\Re\ ••• e\Rer

\enRe 1 enRen .

Proof. Since R is semiregular, so are all e{Rej from [8, Corollary 2.3]. So the proof is complete by Corollary 2.3. □

Recall that a Morita context denoted by (A, B , M, N, ■0,4>) consists of two rings A, B, two bimodules aN b,bM a and a pair of bimodule homomorphisms (called pairings) ip : N ® B M —► A and </> : M N —► B which satisfy the following associativity: ^(n, m)n' = n(f)(m, n'), (j)(m,n)m' = mil)(n,m') for any m, m' € M, n, n' e N. These conditions insure that the set T of generalized matrices ( ^ £); a E A, b e B, m £ M , n e N forms a ring, called the ring of the context. In [7], A. Haghany investigated hopficity and co-hopficity for Morita contexts with zero pairings. Now we study semiregularity for such Morita contexts.


Theorem 2.5. Let T be the ring of a Morita context (A, B, M, N, tp, (ft) with zero pairings. Then T is semiregular if and only if so are A and B.

Proof. Suppose that T is semiregular. Set e = diag(l, 0). Then eTe and (diag(l, 1)- e )r (d ia g ( l , l ) -e ) are both semiregular. Clearly, eTe = diag(A, 0) and (1 —e)T (l — e) = diag(0, B). We directly verify that A and B are both semiregular rings.

Conversely, assume that A and B are semiregular. Choose e\ = diag(l, 0) and e2 — diag(0,1). Then e\Te\ = diag(^4,0) and e2Te2 = diag(0, B). In as much as A and B are semiregular, we know that e\Te\ and e2Te2 are semiregular.

Since T is a Morita context with zero pairings, one checks that J(T) = ̂ )Clearly, e\Te2 = ( g ft ) , e2Te i = ( m o )• Given any ( g g ) G e { I e 2, we have

0 n\ /0 n\ /0 0\ /0 n0 0 ) ~ l o o j l o o j VO 0

(o o ) S 61 ( JM ) 3 (B )) 62 = e‘ J(r )e2-Hence e{T e2 is semiregular. Likewise, e2Te\ is semiregular. It follows by Theorem2.2 that T is semiregular. □

Recall that a right i?-module M is quasi-injective provided that any homomorphism of a submodule of M into M extends to an endomorphism of M. K.R. Good- earl proved that if R is quasi-injective as a right i?-module if R/J(R) is right self- injective, regular ring and idempotents lift modulo J(i?) (see [K.R. Goodearl, Direct sum properties of quasi-injective modules, Bull. Amer. Math. Soc., 82(1976), 108- 110]). For Morita contexts over rings which are right quasi-injective, we can derive the following.

Corollary 2.6. Let T be the ring of a Morita context {A, B, M, N , ip, </>) with zero pairings. If A and B are quasi-injective as right modules, then T is semiregular.

Proof. Clearly, A and B are both semiregular. Therefore we complete the proof by Theorem 2.5. □

If M is a i?-i?-bimodule, then the trivial extension of R by M is the ring R E] M with the usual addition and multiplication defined by (r*i, m i){r2, m2) = {r\r2,r im 2 + m ir2) for r i ,r 2 € R and mi, m2 G M. Now we generalize Theorem 2.5 to module extensions and provide a large class of semiregular rings.

Lemma 2.7. Let R be an associative ring with identity, M a R-R-bimodule. Then J(R G3 M ) = {(x, m) | x G J(it!), m G M }.

Proof. If we write S ~ R M M and J = J(R) then S/(J K M ) = R/J via the map (r,m) h-» r + J from S —> R/J. Hence J(5) C J M M. But if a G J then (1,0) — (a, m) = (1 — a, m) is a unit in S with inverse ((1 — a)-1 , —(1 — a)- 1m (l — a)-1 ), and it follows that J K M C J(5). □

Theorem 2.8. Let R be an associative ring with identity, M a R-R-bimodule. Then the following are equivalent:(1) R is semiregular.

(2) R ^ M is semiregular.


Proof. (1) =>• (2) By virtue of Lemma 2.8, R K M/3(R K M ) = R/J(R). Hence RM M /3(RM M ) is regular. Given an idempotent (e,m) + J (J?§M ) G R ^ M , we see that (e2, era + rae) G J(RMM). Using Lemma 2.8 again, we have e — e2 6 J(i?). Inasmuch as R is semiregular, we can find some / = / 2 G R such that e + 3(R) = f+ 3 (R ). One easily checks that (e ,m )+ J (i?S M ) = ( /,0 ) + J (i?SM ). In addition,

0)2 = ( / , 0) G i? M. Therefore i? Kl M is semiregular.

(2) => (1) Because i? Kl M /J(i? M ) = R/J(R), R/J(R) is regular. Given any idempotent e + J(i2) G R/J(R), there is an idempotent (e ,0) + 3 ( R M ) G R ^M /J(R M M ). So we have an idempotent ( f ,m ) e R M M such that (e, 0) + J(R $ M ) — ( /, m) + J(-R M M ). By Lemma 2.7, we know that e —/ G J(R). In addition, one verifies that / = f 2. Therefore the result follows. □

Corollary 2.9. Let R be an associative ring with identity. Then R is semiregular if and only if so is RM R.

Proof. It is an immediate consequence of Theorem 2.8. □

3. Unit Semiregularity

A ring R has stable range one provided that aR + bR = R implies a + by G U(R) for a y G R. It is well known that R has stable range one if and only if for all finitely generated projective right i?-modules A , B and C, A ® B = A ® C implies B = C. Moreover, we have Ki (R) = U(R)/W(R) if R has stable range one, where W (R) denotes the subgroup of U(R) generated by {p(a, 6, c)p(c, b, a)-1 | p(a, b, c) G U(R ),a ,b ,c G -R}. A semiregular ring R is said to be unit semiregular provided that it has stable range one. In this section, we investigate unit semiregular rings and extend Theorem 2.2 to unit semiregularity.

Lemma 3.1. Let R be a semiregular ring. Then the following are equivalent:(1) R is unit semiregular.

(2) Whenever eR = fR with idempotents e, / G R, (1 — e)R = (1 — f)R .

(3) Whenever eR = fR with idempotents e, / G R, there exists a u G U(R) such that e = u fu~x.

(4) For all finitely generated projective right R-modules A, B and C, A®B = A(&C implies B = C.

Proof. Since R is semiregular, from [1, Corollary 2.3], it is an exchange ring. Thus, we complete the proof by [11, Theorem 9]. □

Lemma 3.2. Let J and K be two-sided ideals in a regular ring R such that JK = 0 and R/K is unit semiregular. Let A and B be finitely generated projective right R-modules.(a) If A/AJ = B/BJ and A/AK = B K , then A = B.

(b) If A/AJ <® B/BJ and A/AK <® B/BK, then A <® B.

Proof. (a) Obviously, R is an exchange ring. Since A/AJ = B/BJ , by an argument of P. Ara et al., we have decompositions A = A\ © A2 , B = Bi © B2 such that Ai = B\, A 2 — A2J and B2 — B2J. From JK - 0, we deduce that


(A\/A\K) (&A2 = A/AK = B/BK = (B\/B\K) © B2. Since R/K is unit-regular, by using Lemma 3.1, we see that A2 = B2. Thus, A = A\ ® A2 = B\ © B2 = B.

(b) is proved in the same manner. □

Now we extend [5, Theorem 4.19] to unit semiregular rings.

Theorem 3.3. Let A and B be finitely generated projective right modules over a unit semiregular ring R.(a) If A/AP = B/BP for all prime ideals P of R, then A = B.

(b) If A/AP <® B/BP for all prime ideals P of R, then A <® B.

Proof. (a) Assume that A ^ B. Set Q = {Q \ Q is an ideal of R such that A/AQ 2= B/BQ}. Clearly, Cl ^ i/j. Choose a positive integer n and idempotent matrices e, / G Mn(R) such that e{nR ) = A and f(n R ) = B. Suppose that Q i Q Q2 Q • • • Q Q m Q •" in fi. Set M = U* Qi. Then M is an ideal of R. If M £ Q, then A/AM = B/BM. So we have matrices g,h G Mn(R) such that e = gh, f = hg, g = egf, h = fhg (mod M ). Therefore we have some i such that e = gh, f = hg, g = egf, h = fhg (mod Qi). This contradicts the choice of Qi. Hence M G £1. That is, is inductive. So R is an ideal of Q of R which is maximal with respect to the property A/AQ ^ B/BQ. Hence Q is not prime, and then we have ideals J D Q and K D Q such that JK C Q. This means that A/AJ = A/AK and B/BJ = B/BK. By Lemma 3.2, we deduce that A/AQ = B/BQ , a contradiction. Therefore we have A = B.

(b) is obtained by a similar route. □

The following result is a direct consequence of Theorem 3.3.

Corollary 3.4. Let A be a finitely generated projective right R-module over a unit semiregular ring R, and let n be a positive integer. If A/AP can be generated by n elements for all prime ideals P of R, then A can be generated by n elements.

Let e, / be idempotents of a semiregular ring R. Call u G eR f right invertible if there exists au G fR e such that uv = e. The ring eR f is said to be unit regular if whenever ax + b = e with a G eR f, x G fR e and b G eRe, there exists a y G eR f such that a + by G eR f is right invertible. We now extend Theorem 2.2 to unit semiregularity as follows.

Theorem 3.5. The following are equivalent:(1) R is unit semiregular.

(2) There exists a complete orthogonal set {e i,-- - ,en} of idempotents of R such that all eiRej are unit semiregular.

Proof. (1) => (2) is trivial.(2) =» (1) By Theorem 2.2, R is semiregular. Since e\Re\, • • • ,enRen all have

stable range one. It follows by [10, Theorem 2.1] that End^(eii? © • • • © enR) also have stable range one. Prom R = e\R © • • • © enR, we conclude that R is unit semiregular. □


Corollary 3.6. Let ei, • • • , en be idempotents of a unit semiregular ring R. Then the ring

/ e\Rei ■ • • ei Ren\

y6jji?6j • • • enRen Jis unit semiregular.

/ e iR ei ■■■ e\R en \

Proof. Let T = I : •. : ). By virtue of Corollary 2.3, T is semiregular.\ e nR e i ••• enR en /

Choose fi = diag(ei, 0, •••,0), •••,/„ = diag(0, • • • , 0, en). Then { / i , — , / „ } is a complete orthogonal set of idempotents of T. Since R has stable range one, one easily verifies that all fiR fi has stable range one as well. Therefore we get the result by Theorem 3.5. □

For unit semiregularity of Morita contexts with zero pairings, we can derive the following.

Proposition 3.7. Let T be the ring of a Morita context (A , B, M, N, iJj, (j)) with zero pairings. Then T is unit semiregular if and only if so are A and B.

Proof. By Theorem 2.5 and [10, Theorem 2.1], the proof is complete. □

Theorem 3.8. Let R be an associative ring with identity, M a R-R-bimodule. Then the following are equivalent:(1) R is unit semiregular.

(2) RM M is unit semiregular.

Proof. (1) => (2) According to Lemma 2.8, RMM/J(RMM ) = R/J(R). Since R has stable range one, so R/J(R). Therefore R E3 M/3(R £3 M ) has stable range one. One easily checks that RM M has stable range one. According to Theorem 2.9, R M M is semiregular, as desired.

(2) =>■ (1) is similar to the consideration above. □

Corollary 3.9. Let R be an associative ring with identity. Then R is a unit semiregular ring if and only if so is RM R.

Proof. We easily obtain this result by Theorem 3.8. □

4. Unit 1-Stable Range

Following Goodearl and Menal (see [6]), a ring R is said to satisfy unit 1-stable range provided that aR + bR = R implies a + bu G U(R) for a u G U(i^). It is well known that Ki (R) = U(R)/V(R) if R satisfies unit 1-stable range, where V(R) = {p (a , b)p(b, a)-1 | p{a,b) = 1 + ab G U(i?)}. In [3, Theorem 2.2], the first author showed that if R satisfies unit 1-stable range then so does Mn(R) for any n > 1. Now we investigate semiregular rings satisfying unit 1-stable range and generalize Theorem 2.2 to such semiregular rings.

Lemma 4.1. The following are equivalent:(1) R satisfies unit 1-stable range.


(2) There exists a complete orthogonal set {e i ,- - - ,en} of idempotents such that all eiRei satisfy unit 1-stable range.

Proof. (1) =>■ (2) is trivial.(2) => (1 ) Construct a ring homomorphism

<p : R/ e\Re\

\enRe i

/ eirei ••• eiren \given by <p(r) = I : •. : I. It is well known that <p is a ring isomorphism.

\ Cn'TCi ••• CfiTCn JBy [4, Theorem 5], we show that R satisfies unit 1-stable range. □

Theorem 4.2. The following are equivalent:(1) R is a semiregular ring satisfying unit 1 -stable range.

(2) There exists a complete orthogonal set {e i ,- - - ,en} of idempotents such that all eiRej are semiregular rings satisfying unit 1-stable range.

Proof. (1) => (2) is trivial by choosing n = 1 and e\ — 1.(2) => (1) In view of Theorem 2.2, R is semiregular. Thus we obtain the result

by Theorem 4.2. □

Corollary 4.3. Let T be the ring of a Morita context with zeropairings. Then T is a semiregular ring satisfying unit 1-stable range if and only if so are A and B.

Theorem 4.4. Let R be an associative ring with identity, M a R-R-bimodule. Then the following are equivalent:

(1) R is a semiregular ring satisfying unit 1-stable range.

(2) R ^ M is a semiregular ring satisfying unit 1 -stable range.

Proof. (1) =>• (2) According to Theorem 2.9, R ^ M is a semiregular ring. From Lemma 2.8, we have RM M/i(RM M ) = R/3(R). Thus R§3M/J(RM M ) satisfies unit 1-stable range, as desired.

(2) =£> (1) Clearly, R is semiregular as well. From RM MjJ(RE\ M ) = R/J(R), we know that R/J(R) satisfies unit 1-stable range. Therefore R satisfies unit 1-stable range, as asserted. □

As an consequence of Theorem 4.4, we can derive the following.

Corollary 4.5. Let R be an associative ring with identity. Then R is a semiregular ring satisfying unit 1-stable range if and only if so is R ^ R.

Acknowledgements. The authors would like to thank the referee for his/her helpful comments and suggestions, which simply many proofs and lead to the new version of this paper.


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Huanyin Chen Department of Mathematics Zhejiang Normal University JinhuaZhejiang 321004PEO PLE’S REPUBLIC OF [email protected]

Miaosen Chen Department of Mathematics Zhejiang Normal University JinhuaZhejiang 321004PEO PLE’S REPUBLIC OF CHINA miaosen@mail. j hptt .zj.cn

mailto:[email protected]

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ON SEMIREGULAR RINGS