On image summand coinvariant modules and kernel summand invariant modules

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 $\varphi$ of $P$ such that ${\rm Im} \varphi$ is a direct summand (a small submodule) of $P$, $\varphi (\ker \nu) \subseteq \ker \nu$, where $(P, \nu)$ 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.