Abstract
Let S be an additively idempotent semiring and Mn(S) be the semiring of all nxn matrices over S. We characterize the conditions of when the semiring Mn(S) is congruence-simple provided that the semiring S is either commutative or finite.
We also give a characterization of when the semiring Mn(S) is subdirectly irreducible for S being almost integral (i.e. xy + yx + x = x for all x,y S). In particular, we provide this characterization for the semirings S derived from the pseudo MV-Algebras.