Characterizations of semirings byp-injective and projective semimodules

摘要

In this paper we obtain characterizations of classes of semirings byP-injective and projective rightR-semimodules. We prove that a semiringRis von Neumann regular if and only if each cyclic rightR-semimodule isP-injective. Moreover, a commutative semiring R whose principal ideals arek-closed is von Neumann regular if and only if every simpleR-semimodule is PP-injective. We also examine some properties of rightPP-semirings, that is, semirings all of whose principal right ideals are projective. It is shown thatRis a rightPP-semiring if and only if the endomorphism semiring of every cyclic projective rightR-semimodule is rightPP.