A 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 concepts related to these modules. Here we introduce the notion of \(n\) -layered \(QTAG\) -modules and discuss some interesting properties of these modules. We show that a \(QTAG\) -module \(M\) is \(n\) -layered if and only if \(M/N\) is an \(n\) -layered module, whenever \(N\) is a finitely generated submodule of \(M\) and \(n\ge 1\) is an integer.

If α denotes the class of all QTAG-modules

A $QTAG$-module $M$ is called almost totally projective if it has a weak nice system. Here we show that the isotype submodules of a totally projective module which are almost totally projective are precisely those that are separable. From this characterization it follows that every balanced submodule of a totally projective module is almost totally projective. Finally, in some special cases we settle the question of whether a direct summand of an almost totally projective module is again almost totally projective.

In the present paper, we consider isotype submodules of those QTAG \mathrm{QTAG} -modules that have countable length and give an in-depth study of their basic and specific properties either in parallel or in contrast to the direct sum of countably generated modules. In addition, we also examine the QTAG \mathrm{QTAG} -modules whose h-pure submodules are direct sums of countably generated modules.