ON EXCHANGE RINGS WITH BOUNDED INDEX OF NILPOTENCE

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1 COMMUNICATIONS IN ALGEBRA, 29(7), (2001) ON EXCHANGE RINGS WITH BOUNDED INDEX OF NILPOTENCE Tongso W Department of Applied Mathematics, Shanghai, Jiaotong University, Shanghai , P. R. China ABSTRACT This paper stdies exchange rings R sch that R=JðRÞ has bonded index of nilpotence. We give several characterizations of sch rings. We prove that if a semiprimitive exchange ring R has index n, then for any maximal two-sided I of R, if R=I has length n, then there exists a central idempotent element e in R sch that ere is an n by n fll matrix ring over some exchange ring with central idempotents, and the restriction p from ere to R=I is srjective. Key Words: Exchange ring; Index of nilpotence; Semiperfect ring Mathematics Sbject Classification: 16D60. 1 INTRODUCTION Recall that the index of a nilpotent element x in a ring R is the least positive integer n sch that x n ¼ 0, while the index of a two-sided ideal I in R is the spremm of the indices of all nilpotent elements of I ([4]). In this note, we stdy exchange rings R sch that R=JðRÞ has bonded index of 3089 Copyright # 2001 by Marcel Dekker, Inc.

2 3090 WU nilpotence. We give several characterizations of sch rings. We prove that R=JðRÞ has index at most n if and only if for any prime ideal P of R containing JðRÞ, the factor ring R=P is semiperfect with weak length at most n, if and only if R contains no set of n þ 1 nonzero orthogonal pairwise-isomorphic idempotent elements. For any semiprimitive exchange ring R with bonded index n and any maximal ideal I of R,ifR=I has index n, then there exists a central idempotent element e in R sch that ere is a n by n matrix ring over some exchange rings with central idempotents and the restriction p : ere! R=I is epimorphic. Recall that a ring R is said to be a (left) exchange ring if the left R-modle R R has the finite exchange property ([8]). This definition is leftright symmetric and the class of exchange rings is closed nder taking factor rings, matrix rings and corners. By [1], this class is also closed nder taking extensions. By [6], a ring R is an exchange ring if and only if idempotents lift modlo every left (right) ideal of R. The class of exchange rings is qite large. It incldes, e.g., von Nemann reglar rings, p-reglar rings ([7]), semiartinian rings ([3]), clean rings ([6]), C*-algebras with real rank zero ([2]). Exchange rings are potent ([5]) in the sense that every one-sided ideal not contained in J ðrþ contains a nonzero idempotent. All rings in this note are associative with identity and all modles are right nital, For any idempotents e and f in R, if er ffi fr as a right R-modle, then we say that e and f are isomorphic. All basic reslts and notions concerning (von Nemann) reglar rings can be fond in [4]. 2 MAIN RESULTS Lemma 2.1. For any right ideal I ofan exchange ring R with I \ JðRÞ ¼0 and any nilpotent element t in End R ðiþ with nilpotent index n, there is a nonzero finitely generated projective direct smmand K of I R with a decomposition K ¼ K 1 K 2 K n sch that K i ffi K j. Proof. Since t n 1 I 6¼ 0 and R is an exchange ring, there is a nonzero idempotent element e of R in t n 1 I. Then we have t n 1 I ¼ er ½t n 1 I \ð1 eþrš, i.e., er is a direct smmand of t n 1 I and tðerþ ¼0. Now consider the splitting epimorphism p 1 t : t n 2 I! t n 1 I! er; where p 1 is the projection from t n 1 I to er. Let t n 2 I ¼ K 1 er L 1, where the restriction of t from K 1 to er is isomorphic. Again consider the splitting epimorphism

3 EXCHANGE RINGS WITH NILPOTENCE 3091 p 2 t : t n 3 I! t n 2 I! K 1 er; where p 2 is the projection from t n 2 I to K 1 er. There is a decomposition t n 3 I ¼ K 31 K 32 K 33 L 2 sch that K 3i ffi K 3j and K 33 ¼ er. Contining this process repeatedly, we finally obtain a direct smmand K of I R sch that K ¼ K n1 K n2 K nn, where K ni ffi K nj and K nn ¼ er and hence, K is a finitely generated projective right R-modle. Lemma 2.2. For any exchange ring R and any nonzero right ideal I ofr with I \ JðRÞ ¼0, ifi has a decomposition I ¼ I 1 I 2 I n where I i ffi I j, then I has a nonzero direct smmand ofthe form K ¼ e 1 R e 2 R e n R; where e i R ffi e j R and e 1 ; e 2 ;...; e n are idempotent elements of R; e i in I i. Proof. Withot loss, assme I 1 ¼ e 1 R and I i ¼ x i R ði 5 2Þ, where 0 6¼ e 1 ¼ e 2 1. For e 1R x 2 R with e 1 R ffi x 2 R, take an nonzero idempotent element e 2 2 x 2 R. Let x 2 R ¼ e 2 R L 2. Then the right R-modle e 1 R has a decomposition e 1 R ¼ K 1 K 2 sch that K 1 ffi e 2 R; K 2 ffi L 2. Let e 1 ¼ e 11 þ e 12, where e 1i 2 K i. Since e 1i 2 e 1 R and e 1 ¼ e 11 e 1 þ e 12 e 1, we have e 1i ¼ e 1 e 1i e 1 2 Re 1. Then e 11 and e 12 are orthogonal idempotent elements of R. In this case, e 11 R e 2 R is a nonzero direct smmand of I 1 I 2 and e 11 R ffi e 2 R. For e 1 R x 2 R x 3 R with e 1 R ffi x 1 R ffi x 2 R, first assme that x 2 is an idempotent element. Take a nonzero idempotent from x 3 R and repeat the above discssions, we obtain the nonzero idempotents e i 2 I i sch that e 1 R e 2 R e 3 R is a direct smmand of I 1 I 2 I 3 and e i R ffi e j R. Contining this process, finally we obtain all the nonzero idempotents e i 2 I i as reqired. We remark that all that is reqired of R in Lemma 2.1 and 2.2 is that R be potent, i.e., every one-sided ideal not contained in JðRÞ contains a nonzero idempotent. This is reminiscent of the early contribtion of Professor Levitzki [5]. In particlar, Levitzki [5, Theorem 3.4] proved that for any semiprimitive potent ring R whose primitive factor rings are artinian, each nonzero two-sided ideal of R contains a central idempotent of R. Lemma 2.3. For any exchange ring R and any nonzero right ideal I ofr with I \ JðRÞ ¼0; ifr has one in its stable range and I has a decomposition I ¼ I 1 I 2 I n where I i ffi I j, then I has a nonzero direct smmand ofthe form K ¼ e 1 R e 2 R e n R, where e i R ffi e j R and e 1 ; e 2 ;...; e n are orthogonal idempotent elements ofr in I. In this case; K is generated by an idempotent element. Proof. By Lemma 2.2, let K be a direct smmand of I with the form K ¼ f 1 R f 2 R f n R, where f i R ffi f j R and all f i are idempotent elements of R in I i. We now prove the lemma by indctions on n.

4 3092 WU When n ¼ 2, we have K ¼ f 1 R f 2 R ¼ f 1 R H 1, where H 1 ¼ ð1 f 1 ÞR \ K. Let e 2 be any nonzero idempotent of H 1. Then e 2 R is a direct smmand of H 1. Since R has one in its stable range, we have H 1 ffi f 2 R ffi f 1 R. Ths we have a decomposition f 1 R ¼ K 1 K 2 sch that K 1 ffi e 2 R. Now let f 1 ¼ t 1 þ t 2, where t i 2 K i and f 1 t i ¼ t i ¼ t i f 1. Then t 1 and t 2 are orthogonal idempotents and e 2 R ffi K 1 ¼ t 1 R. We have t 1 e 2 ¼ 0. Now if e 2 t 1 ¼ 0, then we are done. If e 2 t 1 6¼ 0, then e 2 e 2 t 1 6¼ 0. In this case, e 2 e 2 t 1 is a nonzero idempotent element of K and it is orthogonal with t 1. Since e 2 R ffi t 1 R and ðe 2 e 2 t 1 ÞR e 2 R; t 1 R has a decomposition t 1 R ¼ L 1 L 2 sch that L 1 ffiðe 2 e 2 t 1 ÞR. Let t 1 ¼ h 1 þ h 2 be an orthogonal decomposition of t 1 sch that h i 2 L i ; h 1 R ffiðe 2 e 2 t 1 ÞR and t 1 h 1 ¼ h 1 ¼ h 1 t 1. Then h 1 R is a direct smmand of f 1 R, while ðe 2 e 2 t 1 ÞR is a direct smmand of H 1. In this case, h 1 R ffiðe 2 e 2 t 1 ÞR, and h 1 ; e 2 e 2 t 1 are orthogonal idempotents of K. Now let K ¼ f 1 R f 2 R f n 1 R f n R, where f i R ffi f j R and f 1 ; f 2 ;...; f n are idempotents of K. Sppose that the reslt holds for the n 1 case, i.e., for L ¼ f 1 R f 2 R f n 1 R, we already have idempotents e i 2 L ði ¼ 1; 2;...; n 1Þ sch that e 1 2 f 1 R and e 1 ; e 2 ;...; e n 1 are nonzero orthogonal pairwise-isomorphic idempotents. Let e ¼ e 1 þ e 2 þþe n 1. Then e is an idempotent of K sch that K has a direct smmand H ¼ er e n R, where e n is an idempotent of f n R and e 1 R ffi e n R. Now consider H ¼ er e n R ¼ er N where N ¼½ð1 eþr \ HŠ. We have N ffi e 1 R. Take a nonzero idempotent t n 2 N.Then we have e i t n ¼ 0 for all i ¼ 1; 2;...; n 1. Let t ¼ t n t n e 1 t n e 2 t n e n 1. Then t is a nonzero idempotent in t n R and it is orthogonal with all the ei 2 ði ¼ 1; 2;...; e n 1 Þ. Finally, since e i R ffi e j R ffi N for all the i; j ¼ 1; 2;...; n 1, we have got the nonzero orthogonal pairwise-isomorphic idempotents s i 2 H ði ¼ 1; 2;...; nþ sch that s 1 2 f 1 R. This ends the proof of the reslt. Now we have the following reslt which is similar to [4, Theorem 7.2]: Theorem 2.4. Let R be an exchange ring with one in its stable range. Then for any two-sided ideal I of R with I \ JðRÞ ¼0; the following statements are eqivalent: (1) I has a bonded index ofnilpotence at most n; (2) If e 1 ; e 2 ;...; e nþ1 are orthogonal idempotents in I, then e 1 Re 2 R e n Re nþ1 ¼ 0; (3) No direct smmand of I R is a direct sm of n þ 1 pairwise isomorphic nonzero right ideals; (4) For any idempotent element e in I, er is not a direct sm of n þ 1 pairwise isomorphic nonzero right ideals. Proof. ð3þ )ð1þ. This follows immediately from Lemma 2.1.

5 EXCHANGE RINGS WITH NILPOTENCE 3093 ð1þ )ð2þ. If e 1 x 1 e 2 x 2 e n x x e nþ1 6¼ 0, then for x ¼ e 1 x 1 e 2 þ e 2 x 2 e 3 þþe n x n e nþ1, one has x n ¼ e 1 x 1 e 2 x 2 e n x x e nþ1 6¼ 0 and x nþ1 ¼ 0. Ths I has bonded index k of nilpotence with k 5 n þ 1. ð2þ )ð4þ. Let I ¼ er and assme that the condition (4) fails. Then withot loss, we can assme I ¼ I 1 I 2 I nþ1, where I i ffi I j for all i; j. Let e ¼ e 1 þ e 2 þþe nþ1, where ee i e ¼ e i are elements of I i. Then it is rotine to verify that e 1 ; e 2 ;...; e nþ1 are orthogonal idempotents of I and I i ¼ e i R. Since e i R ffi e iþ1 R, there exist x i 2 e i Re iþ1 and y i 2 e iþ1 Re i sch that x i y i ¼ e i and y i x i ¼ e iþ1. In this case, 0 6¼ x 1 x 2 x n 2 e 1 Re 2 e n Re nþ1. ð4þ )ð3þ. This follows from Lemma 2.3. We remark that the stable range one condition is applied only to proving the implication of ð4þ )ð3þ. We still do not known if this condition cold be dropped. On the other hand, we already known that exchange rings R sch that R=JðRÞ has bonded index of nilpotence have one in their stable range. Since contably many orthogonal idempotents can be orthogonally lifted modlo every two-sided ideal of an exchange ring ([9, Theorem 2.1]), from the proof of Theorem 2.4, we have the following: Corollary 2.5. For any exchange ring R, the following statements are eqivalent: (1) R=JðRÞ has a bonded index ofnilpotence at most n; (2) If e 1 ; e 2 ;...; e nþ1 are orthogonal idempotents, then e 1 Re 2 R e n Re nþ1 JðRÞ; (3) For any idempotent element e in I, er is not a direct sm of n þ 1 pairwise-isomorphic nonzero right ideals of R=J ðrþ. Lemma 2.6 ð½10; Lemma 2:9ŠÞ: For any finitely generated projective modles P; Q over an exchange ring R and any two-sided ideal I ofr; if P=ðPI ÞffiQ=ðQI Þ as right R=I-modles; then we have decompositions P ¼ P 1 P 2 ; Q ¼ Q 1 Q 2 ; sch that P 1 ffi Q 1 ; P 2 ¼ P 2 I and Q 2 ¼ Q 2 I. Lemma 2.7. (1) For any semiprimitive two-sided ideal I ofany exchange ring R with bonded index ofnilpotence ofat most n; the factor ring R=I also has bonded index ofnilpotence ofat most n. (2) For any semiprimitive exchange ring R with one in its stable range and any ideal I ofr, ifboth R=I and I has bonded index at most n, then R also has bonded index at most n.

6 3094 WU Proof. (1) Sppose that there is a nilpotent element t 2 R=I which has an index k 5 n þ 1. Since R=I is semiprimitive, by Theorem 2.4 there exist orthogonal pairwise-isomorphic idempotents f i 2 R=I ði ¼ 1; 2;...; kþ. By [9, Theorem 2.1], we can lift these f i to orthogonal idempotents e 1 ; e 2 ;...; e k 2 R. Since e 1 R ffi e 2 R, by Lemma 2.6 we have decompositions e 1 R ¼ e 11 R e 12 R and e 2 R ¼ e 21 R e 22 R sch that e 11 R ffi e 21 R while e i2 I ¼ e i2 R ði ¼ 1; 2Þ. In this case, e 11 and e 21 are nonzero orthogonal idempotents since e 11 ¼ e 1 e 11 e 1 and e 21 ¼ e 2 e 21 e 2. Still, we have e 11 R ffi e 3 R. Since e 11 R ffi e 3 R and e 11 R ffi e 21 R, we have nonzero orthogonal isomorphic idempotents e i3 2 e i R sch that e i3 R ¼ e i R. Repeating this procedre, finally we obtain nonzero orthogonal pairwise-isomorphic idempotents e ik 2 R ði ¼ 1; 2;...; kþ. In this case the modle K ¼ e 1k R e 2k R e kk R is a direct smmand of R and from it one can easily define a nilpotent element with index k, where k 5 n þ 1, a contradiction. (2) For any semiprimitive exchange ring R and any ideal I of R, sppose that there is a nilpotent element x 2 R with index k ðk 5 n þ 1Þ. By Lemma 2.2 and Lemma 2.3, there are nonzero orthogonal pairwise-isomorphic idempotents e i 2 R ði ¼ 1; 2;...; kþ. In this case, we have orthogonal isomorphic idempotents e i ði ¼ 1; 2;...; kþ in R=I. In this case, if the e 1 6¼ 0, then by Theorem 2.4, we know that R=I has index of nilpotence not less than k 5 n þ 1;ife 1 ¼ 0, then again by Theorem 2.4, the two-sided ideal I has index of nilpotence not less than n þ 1. In either case, there is a contradiction with the assmption. For any semiperfect ring R, ifr=jðrþ ffim n1 ðd 1 ÞM n2 ðd 2 Þ M nr ðd r Þ and n ¼ maxfn i j i ¼ 1; 2;...; rg for some skew fields D i, then we say that R has weak length n. Theorem 2.8. For any exchange ring R, the following statements are eqivalent: (1) R=JðRÞ has a bonded index ofnilpotence at most n; (2) For any prime ideal P ofr containing JðRÞ; R=P is a semiperfect ring with weak length not more than n; (3) R contains no sbset consisting of n þ 1 nonzero orthogonal pairwise-isomorphic idempotent elements. Proof. ð1þ )ð2þ. Sppose that R has bonded index at most n and P is a prime ideal of R with JðRÞ P. Then by [9, Theorem 2.3], R=P is semiperfect. Let J ðr=pþ ¼Q=P. Then R=Q is a semisimple ring and by Lemma 2.7, R=Q has bonded index of at most n. Ths R=P has a weak length at most n,

7 EXCHANGE RINGS WITH NILPOTENCE 3095 ð2þ )ð1þ. Sppose that (2) holds for the exchange ring R. Then for any right primitive ideal I of R,wehaveR=I ffi M k ðd k Þ, where D k is a skew field and n 5 k. Hence R=I has bonded index at most n for all right primitive two-sided ideals I of R. In this case, R=JðRÞ has bonded index at most n since it is a sbdirect prodct of all sch R=I. ð1þ )ð3þ. This follows from Theorem 2.4 ð3þ )ð1þ. If the condition (1) fails, then by Corollary 2.5, we have nonzero orthogonal pairwise-isomorphic idempotents e 1 ; e 2 ;...; e nþ1 in R=JðRÞ. By [9, Theorem 2.1] and Lemma 2.6, we can lift them to nonzero orthogonal pairwise-isomorphic idempotents e 1 ; e 2 ;...; e nþ1 in R, which contradicts with the assmption (3). We remark that the condition (1) in Theorem 2.8 is mch weaker than the following condition: R has bonded index at most n. An obvios example is provided by the local ring Z=ð2 n Þ. Corollary 2.9. Let R be any exchange ring. For any finitely generated projective right R-modle P; let T ¼ End R ðpþ. Then T=JðTÞ has bonded index at most n ifand only ifp has no smmand which is a direct sm ofn þ 1 nonzero pairwise-isomorphic sbmodles. Proof. Since T=JðTÞ ffiend R=JðRÞ ðp=pjþ and PJ is a sperflos sbmodle of P, we can assme J ðrþ ¼0. By Theorem 2.8, T =J ðt Þ has bonded index at most n if and only if T contains no sbset consisting of n þ 1 nonzero orthogonal pairwise-isomorphic idempotent elements. It is obvios that there is a one to one correspondence between the finite set of orthogonal idempotents of T and the direct sm decomposition of the right R-modle P. Also for any (orthogonal) idempotents e; f 2 T; e ffi f if and only if ep R ffi fp R. Ths the reslt hold. We remark that for any exchange ring R, all right primitive factor rings of R have a same bonded index n if and only if R=JðRÞ ffim n ðsþ, where S is a J-semisimple exchange ring with central idempotents. (The proof is similar to that of [4, Theorem 7.14].) Theorem For any semiprimitive exchange ring R with bonded index n and any maximal ideal I of R; if R=I has index n; then there exists a central idempotent element e in R sch that ere is a n by n matrix ring over some exchange ring with central idempotents and the restriction p : ere! R=I is srjective. Proof. Let I 2 MaxSpecðRÞ and R=I ffi M n ðd k Þ, where D n is a skew field. Then choose a set of nonzero orthogonal pairwise-isomorphic idempotents f i 2 R=I ði ¼ 1; 2;...; nþ sch that f 1 þ f 2 þþf n ¼ 1. By the proof of

8 3096 WU Lemma 2.7, this set can be lifted to orthogonal pairwise-isomorphic idempotents e i 2 R ði ¼ 1; 2;...; nþ. Let e ¼ e 1 þ e 2 þþe n. Then we mst have e i Rð1 eþ ¼0 for all i ¼ 1; 2;...; n, since otherwise we can assme that Hom R ðð1 eþr; e 1 RÞ6¼0. In this case, let f : ð1 eþr! e 1 R be any nonzero R-map and let e 11 6¼ 0 be any idempotent in imðfþ. Then e 11 R is a direct smmand of e 1 R and we have a splitting epimorphism pf : ð1 eþr! e 11 R. Hence there is an idempotent e 1nþ1 ¼ð1 eþe 1nþ1 ð1 eþ 2ð1 eþr sch that e 11 R ffi e 1nþ1 R. Since e 11 ¼ e 1 e 11 e 1, we know that e 11 and e 1nþ1 are orthogonal. Finally, since e i 2 R ði ¼ 1; 2;...; nþ are orthogonal pairwiseisomorphic idempotents, we obtain a set of n þ 1 nonzero orthogonal pairwise-isomorphic idempotents e 1i 2 R 2 ði ¼ 1; 2;...; n; n þ 1Þ. By Theorem 2.8, this is impossible. Hence we have erð1 eþ ¼0. Since R is semiprime, e is a central idempotent element of R. Since e 1 R ffi e i R, there exists t i 2 e 1 Re i and s i 2 e i Re 1 sch that t i s i ¼ e 1 ; s i t i ¼ e i. In this case, let f ij ¼ s i t j for all n 5 i; j 5 1. Then we have f 11 þ f 22 þþf nn ¼ e and f ij f pq ¼ d jp s i t q ¼ d jp f iq Ths f ij is a set of matrix nits of the ring ere and ere is isomorphic (as a ring) to M n ðe 1 Re 1 Þ. Now we prove that all idempotents of e 1 Re 1 are central in e 1 Re 1. Let S ¼ e 1 Re 1. For any idempotent element f 2 S, iffsðe 1 f Þ6¼0, then we have frðe 1 f Þ6¼0 and ths, Hom R ððe 1 f ÞR; frþ6¼0. Since JðRÞ ¼0, this again will indces a set of orthogonal pairwise-isomorphic idempotents f i 2 R ði ¼ 1; 2;...; n þ 1Þ, a contradiction. Hence, f is a central idempotent element of S since JðSÞ ¼0. Finally, since 1 e 2 I, wehaveð1 eþrð1 eþ I. Ths the map p : ere! R=I is epimorphic. This completes the proof. Corollary For any exchange ring R sch that R=J ðrþ has bonded index ofnilpotence at most n, ifr has only a finite nmber ofmaximal twosided ideals, then R is semiperfect. Proof. Let S ¼ R=JðRÞ. By Theorem 2.10, there exist a complete set of central orthogonal idempotents e i 2 Sði ¼ 1; 2;...; kþ sch that e i Se i ffi M ni ðs i Þ, where S i are some exchange rings with central idempotents and n 5 n i. Since R has only a finite nmber of maximal ideals, each S i is semisimple. Ths R is semiperfect. Lemma For any projective modle P with the finite exchange property; if P ¼ M 1 þ M 2 þþm n ; then there are sbmodles P i M i sch that P ¼ P 1 P 2 P n. Proof. This is contained in [6, Proposition 2.11]. We end this paper by inclding the following reslt abot exchange rings:

9 EXCHANGE RINGS WITH NILPOTENCE 3097 Proposition For any exchange ring R and any sbspace X of SpecðRÞ containing MaxSpecðRÞ; the clopen sets ofx are exactly those ofthe form UðeÞ; where e is an idempotent element ofr sch that for any I 2 X ; either e 2 I or 1 e 2 I. Proof. For any clopen set W of X, let W ¼ UðaÞ; X ¼ UðaÞ[UðbÞ and UðaÞ\UðbÞ ¼;, where UðaÞ ¼fP 2 X j a =2 Pg. Then R ¼ RaR þ RbR since X ¼ UðRaR þ RbRÞ. Since R is an exchange ring, by Lemma 2.12 we have idempotents e 2 RaR and f 2 RbR sch that R R ¼ er fr. In this case, we already have UðeÞ UðaÞ. IfUðeÞ 6¼UðaÞ, then there is a P 2 UðaÞ sch that P =2 UðeÞ. In this case, we have 1 e =2 P and hence b =2 P. Ths P 2 UðbÞ, a contradiction. Ths we have W ¼ UðeÞ. For this e and any I 2 X, since X is a disjoint nion of UðeÞ and Uðð1 eþþ, we have either e 2 I or 1 e 2 I. Conversely, for any e ¼ e 2 2 R, obviosly we have X ¼ UðeÞ[ Uðð1 eþþ. In addition, if for any I 2 X, either e 2 I or 1 e 2 I, then the intersection of UðeRÞ and Uðð1 eþrþ is empty. Ths UðeÞ is a closed and open sets of X. ACKNOWLEDGMENTS The athor expresses his gratitde to the referee for his=her carefl reading and sggestions which improved the exposition of the paper. REFERENCES 1. Ara, P. Extensions of Exchange Rings. J. Algebra 1997, 197, Ara, P.; Goodearl, K.R.; O Meara, K.C.; Pardo, E. Separative Cancellation for Projective Modles over Exchange Rings. Israel J. Math. 1998, 105, Baccella, G. Right semiartinian rings are exchange rings, Preprint. 4. Goodearl, K.R. von Nemann Reglar Rings; Pitman: London, Levitzki, J. On the Strctre of Algebraic Algebras and Related Rings. Trans. Amer. Math. Soc. 1953, 74, Nicholson, W.K. Lifting Idempotents and Exchange Rings. Trans. Amer. Math. Soc. 1977, Stock, J. On Rings whose Projective Modles have the Exchange Property. J. Algebra 1986, 103, Warfield, R.B. Exchange Rings and Decompositions of Modles. Math. Ann. 1972, 199,

10 3098 WU 9. W, T. Exchange Rings with Primitive Factor Rings Artinian. Algebra Colloqim 1996, 3 (3), W, T. Some reslts on finitely generated projective modles and exchange rings, Preprint. Received Febrary 2000 Revised Jly 2000