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.