Locally injective modules and locally projective modules. (Q1415012)

A module \(V\) is called locally injective if, for any non-essential submodule \(A\subseteq V\), there exists a nonzero injective submodule \(Q\subseteq V\) such that \(Q\cap A=0\). Dually, a module \(W\) is called locally projective if, for any non-small submodule \(B\subseteq W\), there exists a nonzero projective direct summand \(P\) of \(W\) such that \(P\subseteq B\). The author uses locally injective and locally projective modules to study the total \(\text{Tot}(M,W)\) of two modules \(M\) and \(W\). Here, \(\text{Tot}(M,W)\) is the set of all homomorphisms \(f\) in \(\Hom(M,W)\) such that the composition \(gf\) is an idempotent in \(\text{End}(M)\) for some \(g\) in \(\Hom(W,M)\). Several properties of these modules are obtained. In particular, a module is locally injective if and only if it is the injective envelope of a direct sum of injective modules.