(Received August 2000)

Transcription

1 3W ZEALAND JOURNAL OF MATHEMATICS lume 30 (2001), IDEMPOTENTS IN EXCHANGE RINGS H u a n y i n C h e n (Received August 2000) Abstract. W e investigate idempotents in exchange rings and show that an exchange ring R has stable range one if and only if er = f R with idempotents e, f 6 R implies e = u f u - 1 for some u 6 U(i2). Also we show that a simple exchange ring R has stable range one if and only if there exists k > 1 such that all corners of R satisfy the /c-stable range condition. These are generalizations of the corresponding results of D. Handelman, P. Ara and K.R. Goodearl. An associative ring R is called an exchange ring if for every right i?-module A nd any two decompositions A = M N ie/ Ai, where M r = R r and the index et I is finite, there exist submodules A\ C Ai such that A = M ( ig/ A^). We now that local rings, semiperfect rings, semiregular rings, 7r-regular rings, strongly -regular rings and C*-algebras with real rank one (see [3]) are all exchange rings. Recall that R has stable range one provided that ar + br = R with a,b & R mplies that there exists a y R such that a + by G \J(R). It is well known hat R has stable range one if and only if for any right i?-modules A, B and C, \ B = A C implies B = C. Many authors have studied stable range one :ondition and cancellation of modules such as [1], [3], [6], [8], [10-11], [13-14] and 19-20]. In this paper, we investigate idempotents in exchange rings. For an exchange ring R, we prove that it has stable range one if and only if er = fr with idempotents i, f G R implies e = ufu~x for some u G U(R). Moreover, we show that a simple jxchange ring R has stable range one if and only if R is strongly separative if and )nly if there exists k > 1 such that all corners of R satisfy the /c-stable range condition. These extend the corresponding results of D. Handelman, P. Ara and K.R. Goodearl. Throughout, rings are associative with identities and modules are right R-modules. U(i2) denotes the set of all units of R. Call x G R is regular if x = xyx for a y G R and x G R is unit-regular if x = xux for a u G U(-R). Let e be an idempotent of R. We call the ring ere a corner of R. Lemma 1. Suppose ip : er = fr with e = e2, f = f 2 G R. Then ip(e) is regular in R. Proof. Assume that / = ip(er) for some r G R. Then we have / = i[)(e)r, hence 'ip(e)rip(e) = /^ (e). Since tp(e) G fr, we claim that ij)(e) = i/»(e)r?/>(e), as asserted. Theorem 2. Let R be an exchange ring. Then the following are equivalent: 2000 A M S Mathematics Subject Classification: 16E50, 16U99. Key words and phrases: idempotent, exchange ring, stable range condition.

2 1 04 HUANYIN CHEN (2) Whenever ip : er = fr with e = e2, / = f 2 G R, ip(e) is unit-regular. Proof. (2) = (1) Given any regular x G R, then x = xyx and y = yxy for a y G R. Obviously, we have right i?-module isomorphism ip : yxr = yr = xyr given by ip{yr) = xyr for any r G R. So ip(yx) = xyx is unit-regular. In view of [20, Theorem 3], we know that R has stable range one. (1) => (2) By Lemma 1, we have ip(e) is regular. Thus the result follows from [20, Theorem 3]. Corollary 3. Let R be an exchange ring. Then the following are equivalent: (2) Whenever ip : ar = br with regular a,b G R, ip (a) = ua for some u G U (R). Proof. (2) =$ (1) Given ip : er = fr with e = e2, / f 2 G R, then ip(e) = ue for some u G V(R). Thus ip(e) G R is unit-regular. By Theorem 2, we know that R has stable range one. (1) => (2) Given ip : ar = br with regular a,b G R, then we have an idempotent / G br such that br = fr. Clearly, / = ip(ar) = ip(a)r for some r G R. Thus ip(a)rip(a) = ftp {a) = ip (a). Since a G R is regular, in view of [13, Lemma 1], we know that Ra = Rip(a). Thus there exists some u G U(R) such that ip(a) = ua by [6, Corollary 4.5]. Corollary 4. Let R be an exchange ring. Then the following are equivalent: (2) Whenever ip : ar = br with regular a,b G R, R/aR = R/bR. Proof. (2) => (1) Given ip : er = f R with e = e2, / = f 2 G R, then (1 e)r = R/eR = R /fr = (1 f)r. Thus R has stable range one from [19, Theorem 9]. (1) (2) Suppose ip : ar = br with regular a, 6 G R. By Corollary 3, we have u G U(R) such that ip(a) = ua. Thus we have an isomorphism u : R >R given by u(r) = ur for any r G R. Hence there exists (p : R/aR >R/bR such that the following diagrams are commutative: 0» ar» R > R/aR > 0 ip u 1 (p 0 > br R R/bR * 0. It is easy to check that 0 : R/aR >R/bR is also an isomorphism. So we complete the proof. Lemma 5. Whenever ip : er = fr with e = e2, f = f 2 G R and ip(e) is unitregular, e = u fu ~ l for some u G U(i?). Proof. Clearly, we have some r G R such that / = ip{er). Hence / = ip(er) = (fip (e)e)(erf). On the other hand, we show that e = '0-1('0(e)) = ip~x (fip(e)) = - 1(/)^(e) = erip(e) = (erf)(fip (e)e). Set a = e r f, b = fip{e)e. Then e = afb, / = bea, a -- aba and b -- bob. Prom ba -f (1 ba) 1, we see that fip(e)ea + (1 ba) 1, hence fip{e)ea{ 1 / ) + (1 ba){ 1 / ) = 1 - /. Thus fip{e) + (1-6a)(1 - f)ip{e) = (l - fip{e)ea{\ - f))ip{e). Clearly,

3 IDEMPOTENTS IN EXCHANGE RINGS 105 (6 + (1 6a)(l f)ip(e))a + (1 6a)(l (1 f)ip(e)a) = 1. So we claim that (l fip(e)ea{ 1 f))ip(e)a + (1 6a) (1 (1 f)ip{e)a) = 1. We easily check that (l fip(e)ea( 1 f))ip(e) E R is unit-regular because ip(e) is unit-regular and 1 fip{e)ea{ 1 / ) G U(i?). By virtue of [7, Lemma 3 and Corollary 2], we see that a + z(l ba) = v E U(R) for some z E R. Therefore b = bab = b(a+z(l ba))b = bvb. Set u = (1 + a6 i;6)t>(l + ba bv). Obviously, w-1 = (1 6a + 6t!)v_1(l ab + vb). Furthermore, we check that a = eu = u f. Thus e = u fu-1, as required. Theorem 6. Let R be an exchange ring. Then the following are equivalent: (2) Whenever ip : er = fr with e = e2, / = / 2 G R, e = u fu -1 for some ue\j{r). Proof. (1) (2) It follows by Theorem 2 and Lemma 5. (2) => (1) Given ip : er = fr with e = e2, / = f 2 E R, then e = ufu~l for some u E U(R). Hence 1 e = ((1 e)u( 1 /))((1 /)w-1(l e)) and 1 - / = ((1 - /)w 1(l - e) ) ( ( 1 ~ c) (! _ /)) Therefore (1 - e)i? = (1 - /)i2. By [20, Theorem 9], we complete the proof. Corollary 7. Let R be an exchange ring. Then the following are equivalent: (2) Whenever ar = br with regular a,b E R, there exist u,v E U(R) such that a = ubv. Proof. (1) (2) Assume that ip : ar = br with regular a,b E R. Since R has stable range one, we know that a and 6 are both unit-regular. Thus we may assume that a = eu and 6 = fv with e = e2, f = f 2 E R and u,v E U(R). Obviously, er = ar = br = fr. So Re = R f. Therefore Ra = Rb. By Corollary 4, we know that R/aR = R/bR. Likewise, we have R/Ra = R/Rb. In view of [13, Lemma 3], there are u,v E U(R) such that a = ubv. (2) => (1) Given ip : er = fr with e = e2, / = f 2 E R, then we have v,w E U(R) such that e = vfw. In view of [13, Lemma 3], we know that ee~ = u ff^ u - 1 for some u E XJ(R). Therefore e = u fu -1. Thus the result follows by Theorem 6. As an immediate consequence of Corollary 7, we see that a regular ring R is unit-regular if and only if ar = br => a ubv for some u,v E \J(R). Now we give other conditions on idempotents e, / G R under which er = fr implies that there exists some u E U(R) such that e = ufu~x. Proposition 8. Let R be an exchange ring. Whenever ip : er = fr with e = e2, f = f 2 E R and e f = 0, e = u fu -1 for some u E U(R). Proof. Clearly, / = ip(er) for some r E R. Thus / = ip(er) = (fip(e)e)(erf). Similarly to Lemma 5, we claim that e = V;_1(/'0 (e)) = (er/ ) ( / 'J/,(e)e)- Set a = erf, b = fip(e)e. Then e = a / 6, / = 6ea, a = aba and 6 = bab. Since ef = 0, we deduce that 62 = 0. That is, 6 G R is a nilpotent regular element. By [1, Theorem 2], we know that 6 G R is unit-regular. Assume that 6 = bvb for v E U(R). Set u = (1 + a6 vb)v(l + 6a bv). Obviously, u E U(R). Hence e = ufu-1, as asserted.

4 1 06 HUANYIN CHEN Corollary 9. Let R be an exchange ring. Whenever er = (1 e)r with e e2, eue = 0 for some u G U(R). Proof. Suppose er = (1 e)r with e = e2. Since e (l e) = 0, by virtue of Proposition 8, we have e = u( 1 e)u~l for some u e U (R). Therefore eue u( 1 e)e = 0, as required. Following P. Ara et al., we say that an exchange ring R is a separative ring if for any finitely generated projective right i?-modules A and B, A A = A B = B B implies A = B. We call R is a strongly separative ring if for any finitely generated projective right P-modules A and B, A A = A B implies A = B. Separativity and strongly separativity play key roles in the direct sum decomposition theory of exchange rings. It seems rather likely that all exchange rings should be separative (see [3]). Now we investigate idempotents of simple separative exchange rings and simple strongly separative exchange rings. Lemma 10. Let R be a separative exchange ring. Whenever ip : er = fr with e = e2, f = f 2 G R and R(1 e)r R( 1 f)r R, (1 e)r = (1 f)r. Proof. Since R( 1 e)r = R, we can find yi,,ym G R such that 1 G yi(l e)r. Hence R ^2yi(l e)r. So there exists an epimorphism xp : m( 1 e)r > 1 ~ e)r R- Since it* is a projective right i?-module, we know that ip is a split epimorphism. Hence R < m( 1 e)r. Likewise, we have n > 1 such that R < n( 1 f)r. Therefore we show that er < ra( 1 e)r and fr < n(l f)r. As er (1 e)r = R = fr (1 f)r, we have (1 e)r = (1 f)r by [3, Lemma 2.1]. J. Hannah and K.C. O Meara observed that if Rr(a) l(a)r = R( 1 a)r characterizes products of idempotents in R, then R satisfies the following weak unit-regular property for a regular ring R: for all idempotents e, / G R, er = fr and i?(l e)r = i?(l f)r = R (1 e)r = (1 f)r. Lemma 10 shows that separative exchange rings also possess such weak unit-regularity. Theorem 11. Let R be a separative exchange ring. Whenever er = fr with e e2, f = f 2 G R and R( 1 e)r = R( 1 f)r = R, e = u fu ~ x for some ue\j(r ). Proof. Given ip : er = fr with e = e2, / = f 2 G R and R( 1 e)r = i?(l f)r R, then we have an isomorphism (f>: (1 e)r = (1 f)r by Lemma 10. Clearly, there exists some r G R such that / = ip(er). Analogously to the consideration in Lemma 5, we claim that / = (fip(e)e)(erf) and e = (er/)(/^ (e)e). Set a = erf, b = fip(e)e. Then e = afb, f bea, a = aba and b - bob. Now we construct an isomorphism v : ei? (1 e)r > fr (1 f)r given by er + (1 e)s \ >ip(er) + 0((1 e)s) for any r, s G R. For any t & R, we can check that ava(t) = aver f t aip(e)rft aip(er)ft a f(t) = a(t), so a = ava. Set u = (1 ab + av)v~1( 1 ba + va). It is easy to verify that u-1 = (1 + ba va)v( 1 + ab av) and a = u f = eu. Hence e u /u -1, as desired. Corollary 12. Le R be a separative regular ring. Whenever er = f R with idempotents e and f both being products of nilpotents, e = u fu ~ l for some u G U(i?).

5 IDEMPOTENTS IN EXCHANGE RINGS 1 07 Proof. Since e = e2 G R is the product of nilpotents, by [12, Proposition 3.3], we know that R( 1 e)r = i2. Likewise, R{ 1 /)i? = i2. Thus the result follows from Theorem 11. Corollary 13. Let R be an exchange ring satisfying s-comparability. Whenever er = fr with e = e2, / = / 2 G R and R( 1 e)r = i?(l f)r R, e = ufu~l for some u G U (i?). Proof. In view of [15, Theorem 2.2], we see that R is a separative exchange ring. By Theorem 11, we complete the proof. Theorem 14. Let R be a simple exchange ring. Then the following are equivalent: (1) R is separative. (2) Whenever ip : er = fr with nontrivial idempotents e, / G R, e = ufu~l for some u G U(R). (3) Whenever ip : er = / i? with nontrivial idempotents e,/ G i?, (1 e)r = Proof. (1) => (2) Suppose ip : er = fr with nontrivial idempotents e, f R. Since e is a nontrivial idempotent, by the simplity of R, we can find m > 1 such that er < m(l e).r. Similarly, fr < n(l /)i? for some n > 1. Prom ei? (1 e)i? = /i? (1 f)r, we claim that 0 : (1 e)r = (1 f)r by [3, Theorem 2.1]. Analogous to the consideration in Theorem 11, we have an isomorphism v : er (1 e)r >fr (1 - f)r given by er + (1 e)s \ * ip(er) + (p{ (l e)s) for any r, s G i?. Moreover, we have a ava. Set u = (1 ab + av)v~1(l ba + va). Then w-1 = (1 + ba va)v( 1 + ab av) G U (i?) and a = u f = ew, as required. (2) => (3) Suppose ip : er = fr with nontrivial idempotents e, f e R. Then there exists some u G U (i?) such that e = ufu~l. Thus 1 e = ((1 e)u(l /) ) ((1 / ) «-1 (l e)) and 1- / = ( ( l - /) w 1( l - e ) ) ( ( l - e ) u ( l - / ) ), whence (1 e)r = (1 f)r. (3) =r- (1) Given any nonzero regular x G R, then x = xyx and y = yxy for a y G R. Assume that xy ^ 1 and yx ± 1. Set e xy and / = yx. Clearly, we have an isomorphism ip : er = xr = fr given by ip(xr) = yxr for any r G R. Since xy and yx are both nontrivial idempotents of R, we show that 0 : (1 xy)r = (1 yx)r. Now we construct an isomorphism u : er (1 e)r >fr (1 f)r given by u[er + (1 e)s) = ip(er) + 0((1 e)s) for any r,s G R. It is easy to verify that xux{t) = xuex(t) = xip(ex)(t) xyx(t) = x{t) for any t G R, hence x = xux. Therefore we show that every regular in R is one-sided unit-regular, and the proof is completed from [9, Theorem 3]. Corollary 15. Let R be a simple exchange ring. Then the following are equivalent: (2) Whenever tp : er = fr with idempotent e G R and nontrivial idempotent f G R, e = ufu~x for some u G XJ(R).

6 108 HUANYIN CHEN (3) Whenever if) : er = fr with idempotent e G R and nontrivial idempotent f R, (1 - e)r (1 - })R. (4) R is strongly separative. P roof. (1) = (4) is trivial. (4) => (3) Given ip : er = fr with idempotent e G R and nontrivial idempoten / G R, then fr < m(l f)r by the simility of R. Since R is strongly separative, we can derive (1 e)r = (1 f)r from er (1 e)r = f R (1 f)r, as desired. (3) => (2) Given if) : er = fr with idempotent e G R and nontrivial idempoten / G R, then 0 : (1 e)r = (1 f)r. Assume that / = ip(er) for some r G R. Hence / = (fip(e)e)(erf) and e = (erf)(fif)(e)e). Set a = e r f, 6 = fip(e)e. Then e = a / 6, / = bea, a = aba and b = bab. Similarly to the discussion in Theorem 11, we can construct an isomorphism v : er (1 e)r fr (1 f)r such that a = ava. Set u = (1 ab + av) v-1 (l ba + va). Then we show that u G U(i?) and e = ufu-1, as required. (2) => (1) By virtue of Theorem 14, we see that R is a separative exchange rin Assume that xy = 1 and yx ^ 1. Then yx R is nontrivial idempotent. Prom xyi? = yxr, we have 1 rry = w(l yx)w-1 for some u G U(i?). Hence yx = 1, a contradiction. So R is directly finite. Therefore we conclude that R has stable range one by [3, Theorem 3.4]. Following R.B. Warfield ([16]), we call R satisfies the fc-stable range condition if for any (ai,---,ar) G Rr(r > A;), there exist, 6r_i G such that (ai + arbi,, ar_i + arbr- 1) G is unimodular. We denote minjfc i? satisfies the A;-stable range condition and do not satisfy the(fc 1)- stable range condition} by st(r). If there is no such k, we say sr(r) = oo. For further properties of exchange rings satisfying the /c-stable range, we refer readers to [10] and [16-18]. The following result gives a new characterization of strongly separative exchange rings. Lemma 16. Let R be an exchange ring. Then the following are equivalent: (1) R is strongly separative. (2) There exists k > 1 such that all corners of R satisfy the k-stable range condition. Proof. (1)=>(2) Since R is strongly separative, so are all corners of R. Thus all corners of R satisfies the 2-stable range condition by [3, Theorem 3.3], as required. (1)=>(2) Suppose A C = B C with C < A and A,B,C G FP(i?). Applying [5, Lemma 8.5] to V(-R), we have a refinement matrix B C A ( Dx A1 \ c V Bi Ci J such that C\ < A\. Thus we know that 2Ci < Ci Ai = C < A, so A = 2Ci E for a right i?-module E. It is easy to check that A Ci = Di A\ Ci == Di C Di Bi Ci B C i. Therefore Ci A = 3Ci E Ci B with 2Ci < A. By induction, we can find some right.r-module Ck such that Ck A = Ck B with (k + 1 )Ck ^ A. Assume that A = (k -f l)ck D. Then (k + 2)Ck D = Ck B. Since all corners of exchange ring R satisfies the /c-stable

7 IDEMPOTENTS IN EXCHANGE RINGS range condition, so does End^Cfc by the fact that Ck is generated by idempotents. Hence (k + 1 )Ck D = B by [16, Theorem 1.2], so A = B. Therefore R is strongly separative. We know that every direct finite simple exchange ring has stable range one. Furthermore, we see that every simple exchange ring is separative if and only if it has stable range one or satisfies the comparability axiom. Now we extend [2, Theorem 1.2 and Theorem 1.5] to exchange rings and give a classification of simple exchange rings as follows. Theorem 17. Let R be a simple exchange ring. Then the following are equivalent: (2) There exists k > 1 such that all comers of R satisfy the k-stable range condition. Proof. (1)=^(2) Since R has stable range one, so has ere for all idempotents e G R. Hence all corners of R satisfy 1-stable range condition. (2)=^(1) By Lemma 16, we claim that R is strongly separative. Thus the result follows by Corollary 15. Corollary 18. Let R be a simple exchange ring. Then one of the following possibilities occurs: (2) sr(endftp) = oo for all nonzero finitely genertaed projective right R-modules P. (3) sr(endftp) is finite for all finitely generated projective right R-modules P and the set {sr(endftp) P is a finitely generated projective right R-module} is not bound. Proof. It is easy to obtain by Theorem 17 and [2, Lemma 1.3]. Acknowledgments. It is a pleasure to thank the referee for thoroughly reading the article and for helpful suggestions. This research was supported by the National Natural Science Foundation of China (Grant No ) and the Ministry of Education of China. References 1. P. Ara, Stronqlu -K-reqular rinqs have stable ranqe one, Proc. Amer. Math. Soc. 124 (1996), P. Ara and K.R. Goodearl, The almost isomorphism relation for simple regular rings, Publ. Mat. (Barcelona), 36 (1992), P. Ara, K.R. Goodearl, K.C. O Meara and E. Pardo, Separative cancellation for projective modules over exchange rings, Israel J. Math. 105 (1998), P. Ara, K.C. O Meara and D.V. Tyukavkin, Cancellation of projective modules over regular rings with comparability, J. Pure Appl. Algebra, 107 (1996), G. Brookfield, Ph.D. Thesis, University of California at Santa Barbara, M.J. Canfell, Completion of diagrams by automorphisms and Bass first stable range condition, J. Algebra, 176 (1995),

8 1 1 0 HUANYIN CHEN 7. H. Chen, Elements in one-sided unit regular rings, Comm. Algebra, 25 ( H. Chen, Rings with stable range conditions, Comm. Algebra,26 ( H. Chen, Related comparability over exchanqe rinqs, Comm. Algebr 27 (1999), H. Chen, On stable range conditions, Comm. Algebra, 28 (2000), K.R. Goodearl, Von Neumann Regular Rings, Krieger, Malabar, FL, 2nd ed (1979), Pitman, London-San Francisco-Melbourne, J. Hannah and K.C. O Meara, Depth of idempotent-generated subsemigrouj of a regular ring, Proc. London Math. Soc. 59 (1989), R.E. Hartwig and L. Luh, A note on the group structure of unit regular rir elements, Pacific J. Math. 71 (1977), D. Handelman, Perspectivity and cancellation in regular rings, J. Algebn 48 (1977), E. Pardo, Comparability, separativity, and exchange rings, Comm. Algebn 24 (1976), R.B. Warfield, Cancellation of modules and groups and stable range of endc morphism rings, Pacific J. Math. 91 (1980), T. Wu, On the stable range of endomorphism rings of quasi-projective modules Comm. Algebra, 26 (1998), T. Wn and Y. Xu, On the stable range condition of exchange rings, Comm Algebra, 25 (1997), H.P. Yu, Stable range one for exchange rings, J. Pure Appl. Algebra, 98 (1995) H.P. Yu, Stable range one for rings with many idempotents, Trans. Amer. Math Soc. 347 (1995), Huanyin Chen Department of Mathematics Hunan Normal University Changsha P.R. CHINA chyzxl@sparc2.hunnu.edu.cn