$qtag$-modules

In this work we present a systematic study of $n$-layered modules which are closely related to $\Sigma$-modules. For each integer $n \geq 1$ we prove some results for $n$-layered modules concerning when $\Sigma$-modules are direct sum of countably generated modules. Moreover, we discover additional restriction which leads to coinciding of $n$-layered modules and $m$-layered modules for $n>m$.

We define the notion of almost $n$-layered $QTAG$-modules and study their basic properties. One of the main result is that almost 1-layered modules are almost $(omega+1)$-projective exactly when they are almost direct sum of countably generated modules of length less than or equal to $(omega+1)$. Some other characterizations of this new class are also established.