The regularity of munn rings and semigroup rings

Abstract

In this paper, the author studies the regularity of Munn rings and completely 0-simple semigroup rings. When either (i)R is a ring with identity andI and Λ are infinite, or (ii)R is a strong IBN and fully Dedekind-finite ring with identity and eitherI or Λ is finite, in Section 2, the regularity of the Munn ringM(R; I, Λ; P) is characterized; in Section 3, for a completely 0-simple semigroupS =M°(G; I, Λ; P), the regularity ofRS is characterized. Meantime, the author shows that for a locally finite monoidS and a ringR with identity,RS is a strong IBN ring if and only ifR is so;RS is fully Dedekind-finite if and only ifR is so.