Rings defined by $\mathcal{R}$-sets and a characterization of a class of semiperfect rings

Abstract

Introduction. In this paper we give a characterization of semiperfect rings with projective, essential left socle (Theorems 5.1 and 4.4). This characterization is effected through the use of special a-sets. General a-sets are defined and discussed in ?3. Special a-sets are dealt with in ?4. Theorem 5.1 gives necessary and sufficient conditions for a ring with the above properties to be indecomposable, left (or right) perfect, semiprimary, left (or right) artinian, and for the left socle of the ring to be finitely generated. Thus we have, in particular, given a solution to a problem of Goldie [2, p. 268]: very interesting problem is the determination of artinian rings with zero singular ideal. In this connection, see also Gordon [4, Theorem 3.1]. Theorem 5.2 is a special case of Theorem 5.1-the determination of semiperfect rings with projective, essential left socle and a unique isomorphism class of minimal left ideals. The simplest natural instance of Theorem 5.2 is exploited in Theorem 5.3. Here we determine those semiperfect rings with projective, essential left socle in which principal indecomposable left ideals have unique simple submodules. This is a generalization of a theorem of Zaks' who handled the semiprimary case [9, Theorem 1.4, p. 67]. In Proposition 5.5 we give necessary and sufficient conditions for a semiperfect ring R with projective, essential left socle to have projective, essential right socle. This proposition implies that the right socle is typically not even a projective submodule of RR, Our main result in ?2 is the following lemma (Lemma 2.2): If R is a ring in which the identity is a sum of orthogonal idempotents ei, then the radical of R is left T-nilpotent if and only if the radical of eiRet is left T-nilpotent for every i. Also in ?2, we prove what might be a new lemma about reflexive, transitive relations on a finite set (see Lemma 2.7). This lemma allows us to give a normal form for rings of the type characterized in Theorem 5.3. An example (see Remarks 5.4) shows that this normal form simply need not occur in more general cases. We remark that this paper seems to inherently give rise to some apparently hard problems at various levels of abstraction. One glaringly obvious such instance: