Module decompositions by images of fully invariant submodules

Abstract

Let $R$ be a ring with identity, $M$ be a right $R$-module and $F$
be a fully invariant submodule of $M$. The concept of an
$F$-inverse split module $M$ has been investigated recently. In
this paper, we approach to this concept with a different
perspective, that is, we deal with a notion of an $F$-image split
module $M$, and study various properties and obtain some
characterizations of this kind of modules. By means of $F$-image
split modules $M$, we focus on modules $M$ in which fully
invariant submodules $F$ are dual Rickart direct summands. In this
way, we contribute to the notion of a $T$-dual Rickart module $M$
by considering $\overline{Z}^2(M)$ as the fully invariant
submodule $F$ of $M$. We also deal with a notion of relatively
image splitness to investigate direct sums of image split modules.
Some applications of image split modules to rings are given.