On Regular Matrix Semirings

S. Chaopraknoi, K. Savettaseranee, P. Lertwichitsilp

Abstract

A ring R is called a (von Neumann) regular ring if for every $x \in R$; x= xyx for some $y \in R$. It is well-known that for any ring R and any positive integer n, the full matrix ring $M_n(R)$ is regular if and only if R is a regular ring. This paper examines this property on any additively commutative semiring S with zero. The regularity of S is defined analogously. We show that for a positive integer n, if $M_n(S)$ is a regular semiring, then S is a regular semiring but the converse need not be true for n = 2. And for n 3, $M_n(S)$ is a regular semiring if and only if S is a regular ring.