Regularity of Semilattice Sums of Rings

Abstract

Abstract.If R is a supplementary semilattice sum of subrings Ra, a e Cl, then R is regular if and only if each Rat is regular.A ring is said to be regular, in the sense of von Neumann [6], if for each a e R there exists x e R such that axa=a. The concept of (supplementary) semilattice sum is defined in the previous article [8]. In this article, we prove that if R is a supplementary semilattice sum of subrings Rx, a e £2, then R is regular if and only if Rx is regular for every a e £2. We state, without proof, an application of this result to the regularity of semigroup rings.Throughout this paper D will denote a semigroup. Definitions of any concepts not defined herein will be found in [1] or [8].We first prove the main theorem in the case when the semilattice has only two elements.