On image summand coinvariant modules and kernel summand invariant modules

Abstract

In this paper we introduce the concept of im-summand coinvariance and im-small coinvariance; that is, amodule M over a right perfect ring is said to be im-summand (im-small) coinvariant if, for any endomorphism φ of Psuch that Imφ is a direct summand (a small submodule) of P , φ(ker ) ker , where (P; ) is the projective cover ofM. We first give some fundamental properties of im-summand coinvariant modules and im-small coinvariant modules,and we prove that, for modules M and N over a right perfect ring such that N is a small epimorphic image of M, Mis N -im-summand coinvariant if and only if M is (im-coclosed) N -projective. Moreover, we introduce ker-summandinvariance and ker-essential invariance as the dual concept of im-summand coinvariance and im-small coinvariance,respectively, and show that, for modules M and N such that N is isomorphic to an essential submodule of M, M isN -ker-summand invariant if and only if M is (ker-closed) N -injective.