In this paper, we investigate various classes of semirings and complete semirings regarding the property of being ideal-simple, congruence-simple, or both. Among other results, we describe (complete) simple, i.e. simultaneously ideal- and congruence-simple, endomorphism semirings of (complete) idempotent commutative monoids; we show that the concepts of simpleness, congruence-simpleness, and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; we also describe congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of "Morita equivalence" and "simpleness" in the semiring setting, we obtain the following further results: The ideal-simpleness, congruence-simpleness, and simpleness of semirings are Morita invariant properties; a complete description of simple semirings containing the infinite element; the "Double Centralizer Property" representation theorem for simple semirings; a complete description of simple semirings containing a projective minimal one-sided ideal; a characterization of ideal-simple semirings having either an infinite element or a projective minimal one-sided ideal; settling a conjecture and a problem as published by Katsov in 2004 for the classes of simple semirings containing either an infinite element or a projective minimal left (right) ideal, showing, respectively, that semirings of those classes are not perfect and that the concepts of "mono-flatness" and "flatness" for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, lattice-ordered semirings.
The local catenativity of DOL-sequences in free commutative monoids is decidable in the binary case