New characterizations of Von Neumann regular rings and a conjecture of Shamsuddin

Publicacions Matem`atiques, Vol 40 (1996), 383–385. NEW CHARACTERIZATIONS OF VON NEUMANN REGULAR RINGS AND A CONJECTURE OF SHAMSUDDIN Carl Faith Abstract A theorem of Utumi states that if Ris a right self-injective ring such that every maximal ideal has nonzero annihilator, then R modulo the Jacobson radical Jis a finite product of simple rings and is a von Neuman regular ring. We prove two theorems and a conjecture of Shamsuddin that characterize when Ritself is a von Neumann ring, using a splitting theorem of the author on when the maximal regular ideal of a ring splits off. Introduction. Using splitting off theorems for the maximal Von Neumann regular ideal M(R) of a ring Rin the author’s paper [F], we verify a conjecture of Shamsuddin’s as a consequence of the following: Theorem 1. Let Rbe a ring with Jacobson radical J. The f.a.e.c.’s: (1) Ris von Neumann regular (=VNR). (2) Ris semiprime, ¯ R=R/J is VNR, M(R)splits off, and Jis an annihilator (left or right) ideal. As another consequence of M(R) splitting we deduce: Theorem 2. Let Rbe a two-sided continuous (e.g., two-sided selfinjective) ring. Then, Ris VNR iff Ris a semiprime ring whose Jacobson radical is an annihilator ideal. We now state: 384 C. Faith Shamsuddin’s Conjecture. Let Rbe a ring such that R/J is a VNR and a finite product of simple ideals A1,... ,A n. Then Ris VNR iff Ris a semiprime ring and Jis an annihilator ideal. Proof of Theorem 1: The necessity if trivial. Conversely, by assumption, R=M×R2, where M=M(R) and R2has zero regular ideal, that is, M(R2) = 0. Moreover, J⊆R2. We shall show that R2= 0, and then R=M(R) is von Neumann regular. Since Ris semiprime each nonzero ideal Iof Rsatisfies ⊥II=I⊥I=0 where the exponent ⊥denotes the annihilator on the appropriate side. Clearly ⊥J⊇Mhence ⊥J=M⊕(⊥JR2), hence assuming Jis a right annihilator, then (∗)J=( ⊥J)⊥=M⊥(⊥JR2)⊥. Since ⊥JR2=R2(J) is the left annihilator of Jin R2, then ⊥JR2⊥ =R2(J)⊥=M+rR2(J) where R() denotes right annihilation in R2. Using (∗) and the fact that M⊥=R2, then (∗∗)J=M⊥(M+rR2(J)) =R2(M+rR2(J)). Since R2M= 0, and R2⊇rR2(M), then (∗∗) implies that J= rR2(J). If R2= 0, then J=R, hence 2(J)= 0. But R2is semiprime (along with R), hence 2(J)J= 0 which shows that L=2(J) embeds canonically in ¯ R2=R2/J. Since ¯ R=R/J =M×(R2/J) is VNR, then ¯ R2=R2/J is also VNR, and hence Lis a VNR ideal. This shows that Lis an VNR ideal of R2contrary to the fact that M(R2)=0, that is R2has no nonzero regular ideals. This contradiction shows that R2= 0, and hence R=M(R) is VNR as asserted. Proof of Conjecture: By Theorem 2 of [F] (and its Corollary), M= M(R) splits off in Ras a ring direct summand iff the canonical image ¯ M of Min ¯ R=R/J splits off as a right or left ideal. Under the assumption Regular rings and a conjecture of Shamsuddin 385 of the conjecture, the only ideals of ¯ Rare the finite products of a subset of {0,A 1,... ,A n}without repetitions, and all of these are necessarily ring direct summands of ¯ R.Thus ¯ Msplits off in ¯ R, hence Msplits off in R, so Theorem 2 applies to prove that Ris VNR. Proof of Theorem 2: By the main theorem of [F], M(R) splits off in any two-sided continuous (eg. self-injective) ring, so Theorem 1 applies. We conclude with another corollary as applications of two theorems of Utumi. In Corollary 3. If Ris a right self-injective ring in which maximal ideals have nonzero annihilators, then Ris semiprime iff Ris VNR. In this case, Ris a finite product of simple rings. Proof: By Theorem 2.3 of [U1], Ris a finite product of simple rings, and Jis an annihilator ideal. Moreover, R/J is VNR by Theorem 4.8 of [U2]. Thus the truth of Shamsuddin’s conjecture completes the proof. Acknowledgement. Professor Ahmad Shamsuddin made this conjecture during his visit to Rutgers University in September 1994. I have to thank him also for the reference [U1] used in the proof of Corollary 3. References [F] C. Faith, The maximal regular ideal of self-injective and continuous rings splits off, Arch. Math. 4(1985), 511–521. [U1] Y. Utumi, Self-injective rings, J. Algebra 6(1967), 59–64. [U2] Y. Utumi, Continuous rings and self-injective rings, Trans. Amer. Math. Soc. 118 (1965), 158–173. Department of Mathematics Rutgers University Hill Center for Mathematical Sciences Busch Campus New Brunswick, New Jersey 08903 U.S.A. Author’s mailing address: 199 Longview Drive Princeton, NJ 08540 U.S.A. Primera versi´o rebuda el 5 de Desembre de 1995, darrera versi´o rebuda el 10 d’Abril de 1996