TURKISH JOURNAL OF MATHEMATICS, vol.43, no.3, pp.1456-1473, 2019 (Journal Indexed in SCI)
In this paper we introduce the concept of im-summand coinvariance and im-small coinvariance; that is, a module M over a right perfect ring is said to be im-summand (im-small) coinvariant if, for any endomorphism phi of P such that Im phi is a direct summand (a small submodule) of P , phi(ker v) subset of ker v, where (P, v) is the projective cover of M. 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, M is N-im-summand coinvariant if and only if M is (im-coclosed) N-projective. Moreover, we introduce ker-summand invariance 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 is N-ker-summand invariant if and only if M is (ker-closed) N-injective.