The Regularity of Munn Rings and Semigroup Rings

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 and I and A are infinite, or (ii) R is a strong IBN and fully Dedekind-finite ring with identity and either I or A is finite, in Section 2, the regularity of the Munn ring M(R; I, A; P) is characterized; in Section 3, for a completely 0-simple semigroup S M°(G; I, A; P), the regularity of RS is characterized. Meantime, the author shows that for a locally finite monoid S and a ring R with identity, RS is a strong IBN ring if and only if R is so; RS is fully Dedekind-finite if and only if R is so.