www.math.science.cmu.ac.th/thaijournal Online ISSN 1686-0209 On Regular Matrix Semirings

Abstract: A ring R is called a (von Neumann) regular ring if for every x 2 R; x = xyx for some y 2 R. It is well-known that for any ring R and any positive integer n, the full matrix ring Mn(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 dened analogously. We show that for a positive integer n, if Mn(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, Mn(S) is a regular semiring if and only if S is a regular ring