On Regular Matrix Semirings: S. Chaopraknoi, K. Savettaseranee, P. Lertwichitsilp - Team, Support (Thai Journal of Mathematics, 2009)
A ring R is called a (von Neumann) regular ring if for every x R; x= xyx for some y 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.