Modules which are weak extending relative to module classes. (Q5932719)

Two notions of weak extending modules with respect to a class of modules \(\mathcal X\) are introduced and related to other notions of extending relative to a module class. An \(R\)-module \(M\) is called weak type \(1\) \(\mathcal X\)-extending if for every \(\mathcal X\)-submodule \(N\) of \(M\) there exists a complement \(K\) of \(N\) in \(M\) such that \(K\) is a direct summand of \(M\). \(M\) is called weak type \(2\) \(\mathcal X\)-extending if every \(\mathcal X\)-submodule of \(M\) is essential in a direct summand of \(M\). The authors obtain basic properties of such modules similar to known results about extending modules. As one example, it is shown that an \(R\)-module \(M\) is weak type 1 \(\mathcal F\)-extending if and only if \(M=Z_2(M)\oplus M'\) for some type 1 \(\mathcal M\)-extending submodule \(M'\). Here \(\mathcal M\) is the class of all \(R\)-modules, \(\mathcal F\) is the class of nonsingular \(R\)-modules, and \(Z_2(M)\) is the Goldie torsion submodule of \(M\).