On continuous rings and self injective rings

YUZO UTUMI 0. 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. 0.2.Condition.If Sf, f=f2, isisomorphic to a left ideal B, then B also is generated by an idempotent.We call a ring S which satisfies the above two conditions left continuous.Then, if S is left continuous, S / N(S) is a (von Neumann) regular ring which is left continuous in the sense that the lattice of principal left ideals of S is upper continuous.We shall also show some sufficient conditions for a left continuous ring to be left self injective.1. Conditions 1.3, 1.4.1.1.Lemma.Let S be a ring with Condition 0.2, and A a left ideal.Let e and f be idempotents such that Se C(l -f)=0.If Se is an essential extension of A, then Sef is generated by an idempotent, and is an essential extension of Af.Proof.Since Se n S(l -/) = 0, the right multiplication of / gives an isomorphism of Se onto Sef.Hence Sef is an essential extension of Af.By Condition 0.2 Sef is generated by an idempotent.1.2.Theorem.Any left continuous ring satisfies the following two conditions:1.3.Condition.For any idempotent e, and for any left ideal A contained in Se, there exists an idempotent fe Se such that Sfis an essential extension of A.