Almost Quasi-mininjective Modules

Let $M$ be a right $R$-module, $S= \text{End}_R (M).$ The module $M$ is called almost quasi-mininjective (or $AQ$−mininjective ) if, for any simple $M$−cyclic submodule $s(M)$ of $M$, there exists a left ideal $X_s$ of $S$ such that $l_S (Ker(s)) = Ss \oplus X_s$ as left $S$− modules. In this paper, we give some characterizations and properties of $AQ$− mininjective modules.