Generalizations of basic and large submodules of $QTAG$-modules

Abstract

A $QTAG$-module $M$ over an associative ring $R$ with unity is $k$-projective if $H_k(M)=0$ and for a limit ordinal $\sigma ,$ it is $\sigma$-projective if there exists a submodule $N$ bounded by $\sigma$ such that $M/N$ is a direct sum of uniserial modules. $M$ is totally projective if it is $\sigma$-projective for all limit ordinals $\sigma .$ If $\alpha$ denotes the class of all $QTAG$-modules $M$ such that $M/H_\beta (M)$ is totally projective for every ordinal $\beta <\alpha ,$ then these modules are called $\alpha$-modules. Here we study these $\alpha$-modules and generalize the concept of basic submodules as $\alpha$-basic submodules. It is found that every $\alpha$-module $M$ contains an $\alpha$-basic submodule and any two $\alpha$-basic submodules of $M$ are isomorphic. A submodule $L$ of an $\alpha$-module is $\alpha$-large if $M=L+B,$ for any $\alpha$-basic submodule $B$ of $M.$ Many other interesting properties of $\alpha$-basic, $\alpha$-large and $\alpha$-modules are studied.