Subsocles and direct sum of uniserial modules

Abstract

Suppose $M$ is a $QTAG$-module with a subsocle $S$ such that $M/S$ is a direct sum of uniserial modules. Our global aim here is to investigate an interesting connection between the structure of $M/S$ and the $QTAG$-module $M$. Specifically, the condition $S=Soc(N)$ for some $h$-pure submodules $N$ of $M$ allows $M$ to inherit the structure of $M/S$.