| Abstract: | ABSTRACT We study conditions on an ideal A of a self-injective R such that the factor ring R/ A is again self-injective, extending certain of our results for PF rings (Faith, 2006
Faith , C. ( 2006 ). Factor rings of pseudo-Frobenius rings . J. Algebra and Its Applications 6 :(to appear). CSA] Web of Science ®] , Google Scholar]). We also consider the same question for p -injective, and for CS -rings. For the CS -rings we consider conditions under which A splits off as a ring direct factor, equivalently, when A is generated by a central idempotent. Definitive results are obtained for an ideal A which is semiprime as a ring, that is, has no nilpotent ideals except zero, and which is a right annihilator ideal. Then A is said to be an r -semiprime right annulet ideal, and is generated by a central idempotent in the following cases: (1) whenever A is generated by an idempotent as a right (or left) ideal (Theorems 3.4, 3.6); (2) in any Baer ring R (Theorem 3.5); (3) in any right and left CS -ring R (Theorem 4.2), and (4) in any right nonsingular right CS -ring R (Theorem 5.5). These results also generalize results of the author in Faith (1985
Faith , C. ( 1985 ). The maximal regular ideal of self-injective and continuous rings splits off . Arch. Math. 44 : 511 – 521 . CROSSREF] CSA] Crossref], Web of Science ®] , Google Scholar]), where it is proven that the maximal regular ideal M( R) splits off in any right and left continuous ring. The results are applied in Section 6 to extend theorems of Faith (1996
Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . CSA] Crossref], Web of Science ®] , Google Scholar]) characterizing VNR rings, and, as the title of Faith (1996
Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . CSA] Crossref], Web of Science ®] , Google Scholar]) suggests, extend the conjecture of Shamsuddin. |
