The aim of this paper is to investigate quasi-corational, comonoform, copolyform and $\alpha$-(co)atomic modules. It is proved that for an ordinal $\alpha$ a right $R$-module $M$ is $\alpha$-atomic if and only if it is $\alpha$-coatomic. And it is also shown that an $\alpha$-atomic module $M$ is quasi-projective if and only if $M$ is quasi-corationally complete. Some other results are developed.

Let A be a finite-dimensional algebra over an algebraically closed field with radical square zero, and such that all simple A-modules have dimension at most two. We give a characterization of those A that have finitely many conjugacy classes of left ideals.

Let $M$ be a module and $\mu$ be a class of modules in $Mod-R$ which is closed under isomorphisms and submodules. As a generalization of essential submodules Özcan in [8] defines a $\mu$-essential submodule provided it has a non-zero intersection with any non-zero submodule in $\mu$. We define and investigate $\mu$-singular modules. We also introduce $\mu$-extending and weakly $\mu$-extending modules and mainly study weakly $\mu$-extending modules. We give some characterizations of $\mu$-co-H-rings by weakly $\mu$-extending modules. Let $R$...

In this paper we introduce the concept of $\tau$-extending modules by $\tau$-rational submodules and study some properties of such modules. It is shown that the set of all $\tau$-rational left ideals of ${}_{R}R$ is a Gabriel filter. An $R$-module $M$ is called $\tau$-extending if every submodule of $M$ is $\tau$-rational in a direct summand of $M$. It is proved that $M$ is $\tau$-extending if and only if $M=Re{j}_{M}E(R/\tau \left(R\right))\oplus N$, such that $N$ is a $\tau$-extending submodule of $M$. An example is given to show that the direct sum of $\tau$-extending modules need not be $\tau$-extending....