Upper Basic and Congruent Submodules of \(QT AG\)-Modules

Abstract

The cardinality of the minimal generating set of a module \(M\) i.e \(g(M)\) plays a very important role in the study of \(QT AG\)-Modules. Fuchs [1] mentioned the importance of upper and lower basic subgroups of primary groups. A need was felt to generalize these concepts for modules. An upper basic submodule \(B\) of a \(QT AG\)-Module \(M\) reveals much more information about the structure of \(M\). We find that each basic submodule of \(M\) is contained in an upper basic submodule and contains a lower basic submodule. Two submodules \(N, K \subset M\) are congruent if there exists an automorphism of \(M\) which maps \(N\) onto \(K\). In this case \(M/N \cong M/K\)and \(N \cong K\), but these conditions are not sufficient for the congruence. This motivates us to find the sufficiency conditions in terms of Ulm invariants and the extensions of height preserving