ON SEM IREGULAR RINGS

Transcription

1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 32 (23), 11-2 ON SEM IREGULAR 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 22) 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 e R 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. 2 A M S Mathematics Subject Classification: 16E5, 16U99.

2 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 eir ei/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 en 1 G R., en in a ring R is said to be complete Theorem 2.2. Let {ei,,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 = ei(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 {ei,, 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 k \6l^nl6l Cl r ln^n^ Clrnn^-nJ rij G R{ 1 < i, j < n) 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\r ei e i R e n \ enr e 1 en R en / :. ; ) is semiregular.

3 ON SEMIREGULAR RINGS 13 e\re i ei Rer Proof. Set T = ;. ; \ CjiRsn / idempotents g\, /12 T such that / exre^ Suppose that T is semiregular. Then we have \o Clearly, <?ix7i2 is also semiregular. For any x G eir e2-, we have some y E e^rei such that / X o o\ o\ 9 ith2. \ and V / x \ : : ^ / /o y : : \ \ ( X \ Vo V G gi3{t)h,2 (o \ y / / y \ o It is easy to verify that \. / J(T) = J(diag(ei, : VO \ / /o y : : \,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(^r^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 fi = diag(ei,,, ),, f n = diag(,,, en). Then { / i,, / } is a complete orthogonal set of idempotents of T. Consider / i T / 2. eire2 Clearly, f 1T f 2 =.

4 14 HUANYIN CHEN AND MIAOSEN CHEN ( x \ : : : : I G f\t f2, we have a y e e2re\ such that y yxy and - / x xyx eij(i?)e2 because e\re2 is semiregular. Analogously to the consideration above, we verify that ( X \ \ and that V /o X \ / \ y (Q y Vo / ^\ /O * \ N! I e / i J ( T ) / 2,, V> / / y ( ' y \ Therefore / 1T /2 is semiregular 2.2, we get the result. Vo / Likewise, all fit fj are semiregular. Using Theorem We note that the ring in (2) is in fact isomorphic to End#(eii? where e\r enr denotes the external direct sum. enr), Corollary 2.4. Let ei, ring, en be idempotents of a semiregular ring R. Then the ( e\re\ e\rer is semiregular. \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,,4>) consists of two rings A, B, two bimodules an b,b M 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.

5 ON SEMIREGULAR RINGS 15 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, ). Then ete and (diag(l, 1)- e )r (d ia g (l,l)-e ) are both semiregular. Clearly, ete = diag(a, ) and (1 e)t (l e) = diag(, B). We directly verify that A and B are both semiregular rings. Conversely, assume that A and B are semiregular. Choose e\ = diag(l, ) and e2 diag(,1). Then e\te\ = diag(^4,) and e2te2 = diag(, B). In as much as A and B are semiregular, we know that e\te\ and e2t e2 are semiregular. Since T is a Morita context with zero pairings, one checks that J(T) = ^ ) Clearly, e\te2 = ( g ft ), e2t ei = ( m o ) Given any ( g g ) G e { I e 2, we have n\ / n\ / \ / n ) ~ lo oj lo oj VO (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 Theorem 2.2 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. Goodearl proved that if R is quasi-injective as a right i?-module if R/J(R) is right selfinjective, 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), 18-11]). 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,rim 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,) (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.

6 16 HUANYIN CHEN AND MIAOSEN CHEN 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(RM M ). 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 ) = (/, ) + J(i?S M ). In addition, )2 = (/, ) 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,) + 3 (RM) G R ^M /J(R M M ). So we have an idempotent ( f,m ) e R M M such that (e, ) + 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 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 = f R 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 J K = 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 A 2J and B2 B2J. From J K -, we deduce that

7 ON SEMIREGULAR RINGS 17 (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 A 2 = B2. Thus, A = A\ A 2 = 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 = fh g (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 J K 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 a u G fr e such that uv = e. The ring e R f is said to be unit regular if whenever ax + b = e with a G er f, x G fr e and b G er e, there exists a y G er f such that a + by G e R 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 [1, Theorem 2.1] that End^(eii? enr) also have stable range one. Prom R = e\r enr, we conclude that R is unit semiregular.

8 18 HUANYIN CHEN AND MIAOSEN CHEN Corollary 3.6. Let ei,, en be idempotents of a unit semiregular ring R. Then the ring / e\rei ei Ren\ is unit semiregular. y6jji?6j enren J / e ir e i e\r en \ Proof. Let T = I :. : ). By virtue of Corollary 2.3, T is semiregular. \ e n R e i en R en / Choose fi = diag(ei,,,),,/ = diag(,,, 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 [1, 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, RM M /J(RM M ) = 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. semiregular ring if and only if so is RM R. Proof. We easily obtain this result by Theorem 3.8. Then R is a unit 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.

9 ON SEMIREGULAR RINGS 19 (2) There exists a complete orthogonal set {e i,---,e n} 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 J By [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 zero pairings. 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.

10 2 HUANYIN CHEN AND MIAOSEN CHEN References 1. P. Ara, Extensions of exchange rings, J. Algebra, 197 (1997), F.J. Costa-Cano and J.J. Simon, On semiregular infinite matrix rings, Comm. Algebra, 27 (1999), H. Chen, Units, idempotents and stable range conditions, Comm. Algebra, 29 (21), H. Chen, Stable ranges for Morita contexts, SEA Bull. Math. 25 (21), K.R. Goodearl, Von Neumann Regular Rings, Pitman, London-San Francisco- Melbourne, 1979; 2nd ed., Krieger, Malabar, FL, K.R. Goodearl, P. Menal, Stable range one for rings with many units, J. Pure Appl. Algebra, 54 (1988), A. Haghany, Hopficity and co-hopficity for Morita contexts, Comm. Algebra, 27(1999), W.K. Nicholson, Semiregular modules and rings, Can. J. Math. X X V III (1976), W.K Nicholson, Extensions of clean rings, Comm. Algebra, 29 (21), R.B. Warfield, Cancellation of modules and groups and stable range of endomorphism rings, Pacific J. Math. 91 (198), H.P. Yu, Stable range one for exchange rings, J. Pure Appl. Algebra, 98 (1995), Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq. 7 (2), Huanyin Chen Department of Mathematics Zhejiang Normal University Jinhua Zhejiang 3214 PEOPLE S REPUBLIC OF CHINA chyzxl@sparc2.hunnu.edu.cn Miaosen Chen Department of Mathematics Zhejiang Normal University Jinhua Zhejiang 3214 P E O PLE S REPU BLIC OF CHINA miaosen@mail. j hptt.zj.cn