Essentially finitely indecomposable QTAG-Modules

A right module $M$ over an associative ring with unity is a $QTAG$-module if every finitely generated submodule of any homomorphic image of $M$ is a direct sum of uniserial modules. There are many fascinating results related to these modules and essentially indecomposable modules are extensively researched. Motivated by these modules we generalize them as essentially finitely indecomposable modules whose every direct decomposition $M=bigopluslimits_{kin I} M_k$ implies that there exists a positive integer $n$ such that $H_n(M_i)=0$ for all $M_i$'s except for a finite number of $M_i$'s. Here we investigate these modules and their relationship with $HT$-modules. The cases when the modules are not $HT$-modules are especially highlighted.