On some characteristic properties of self-injective rings

Abstract

A ring with unit element is said to be left seif-injective if and only if every (left) R-homomorphism of a left ideal of R into R can be given by the right multiplication of an element of R. In [2], IkedaNakayama introduced the following conditions in a ring R with unit element: (A) Every (left) R-homomorphism of a principal left ideal of R into R may be given by the right multiplication of an element of R. (Ao) Every (left) R-homomorphism of a principal left ideal L of R into a residue module R/L', of R modulo a left ideal L', may be obtained by the right multiplication -of an element, say c, of R: x-*xc (mod L'), (xEL). (B) If I is a finitely generated right ideal in R, then the set of right annihilators of the set of left annihilators of I is I. (B*) If I is a principal right ideal in R, then the set of right annihilators of the set of left annihilators of I is I. We introduce another condition: (C) If F is a finitely generated left free R-module and M is a cyclic submodule of F then any R-homomorphism of M into R can be extended to a R-homomorphism of F into R. In this paper, we shall prove the following: In a ring with 1, (B) holds if and only if (C) holds. If R is a ring with 1 such that every principal left ideal is projective, then the three conditions (A), (AO) and (B) are equivalent. If R is a ring with 1 such that the right singular ideal (refer to [4] for definition) is zero, then R is a semisimple ring with minimum conditions on one-sided ideals if and only if R satisfies the maximum condition for annihilator right ideals and the condition (B). In particular, a regular ring R with 1 is a semisimple ring with minimum conditions on one-sided ideals if and only if it satisfies the maximum condition for annihilator right ideals. In a simple ring R with 1, the condition (B*) and the existence of a maximal annihilator left ideal in R are necessary and sufficient conditions for R to satisfy minimum conditions on one-sided ideals. In a ring with 1, the condition (B*) implies that the left singular ideal of R is, indeed, the Jacobson radical of R. In the sequel, if X is a subset in R, we denote the set of left (right)