Stanley S. Page

Quasi-dual rings

In this paper, all rings R are associative with identity and all modules are unitary right R-modules unless specified otherwise. If every right ideal of a ring R is an annihilator, then R is called a right dual ring. Analogously, one can define left dual rings. A right and left dual ring is called a dual ring. Dual rings were discussed in detail by Hajarnavis and Norton [5] . As a generalization of a right dual ring, a right quasi-dual ring is defined to be a ring for which every right ideal is a direct surnmand of a right annihilator. Right quasi-dual rings can be characterized as those rings R for which every singular cyclic R-module is cogenerated by R. We prove that any right quasi-dual ring R is right Kasch and J(R) = ZI(R), and, if in addition R has ACe on essential right (or left) ideals, J(R) is nilpotent. If R is a quasi-dual ring (see Definition 2.1), then SOC(RR) = Soc(RR) = l(J(R)) = r(J(R)) and J(R) = Zr(R) = ZI(R) = r(Soc(R)) = l(Soc(R)). For a quasi-dual ring R, it is proved that R cogenerates every Goldie torsion right R-module iff Z2(RR) is injective as a right R-module. This result is then used to prove that,

Large fpf rings

The object of this paper is to show that many of the known results concerning the structure of semiperfect FPF rings can be extended to a larger class of FPF rings. The main attributes of this larger class of rings are they have enough principle idempo-tents and idempotents lift modulo the Jacobson radical. We call these rings epi-semiperfect rings.

Images of injectives and universally cotorsionless modules

Kaplansky [2] proved that if P is a projective module, then every f.g. submodule of P is contained in a finitely generated direct summand iff P is the direct sum of f.g. projectives. We show that in order that all injectives have the dual property to the above statement,, for ach pair of simples (S1, S2), Hom(Ŝ1,Ŝ2) must be an Artinian and Noet. i.ian End(Ŝ1) module where i. is the injective hull of Ŝi. This leads to a study of universally cotorsionless modules.