On Continuous Rings and Self Injective Rings

Abstract

YUZO UTUMI0. Throughout this paper we assume that every ring has a unit element.A module is called injective if it is a direct summand of every extension module.A ring is said to be left self injective if it is injective as the left module over itself.The main results we shall show in the present paper are the following : Let S be a left self injective ring.Then S / N(S) is also left self injective, where N(S) denotes the Jacobson radical of S. Any system of orthogonal idempotents of SI N(S) can be lifted to a system of orthogonal idempotents of S.This theorem about orthogonal idempotents can be proved under a somewhat weaker assumption than the left self injectivity of S. In fact, it is enough to suppose that S is a ring satisfying the following two conditions: 0.1.Condition.For any left ideal A there is an idempotent e such that Se is an essential extension of A.