The structure of modules (Q1340022)

The paper starts off with the definition of a maximal quotient projective submodule \(M\) of an \(R\)-module \(X\) as a maximal submodule of \(X\) such that \(X/M\) is projective. The intersection of all maximal quotient projective submodules of \(X\) is called the projective radical of the \(R\)- module \(X\), and denoted \(P(X)\). If \(P(X) = 0\), \(X\) is said to be a subprojective module. Some properties of subprojective modules and the projective radical are derived first and these are used in finding properties of Artinian and semilocal rings. If \(X\) is an \(R\)-module, the semi-reflexive radical of \(X\) is defined as \(S(X) = \bigcap_{f : X\to R} \text{ker }f\). Properties of \(S(X)\) and of semi-reflexive modules are studied. The paper culminates in a characterization of Prüfer domains.