On projective covers of semimodules and perfect semirings

Abstract

We consider two natural concepts of "perfectness" in a semiring context: the perfectness in terms of projective covers, and the perfectness in Bass's sense. For both cases we show that the class of right (left) perfect semirings coincides with the class of right (left) perfect rings; in particular, we completely confirm Conjecture of [Y. Katsov, Toward homological classification of semirings: Serre's conjecture and Bass's perfectness in a semiring context. Algebra Universalis52 (2004) 197–214].