Cofinitely \(F\)-supplemented modules (Q6566488)

Let \(R\) be a unital ring, \(M\) a left \(R\)-module, and \(U,\, V\) and \(F\) submodules of \(M\) with \(F\) proper. \(V\) is an \(F\)-supplement of \(U\) in \(M\) if \(V\) is minimal in the collection of submodules \(F\subseteq X \subset M\) such that \(M = U + X\). The submodule \(U\) has ample \(F\)-supplements in \(M\) if every submodule \(V\) such that \(M = U + V\) contains an \(F\)-supplement of \(U\) in \(M\). The module \(M\) is called (amply) \(F\)-supplemented if every submodule of \(M\) has an (ample) F-supplement in \(M\). \(U\) is cofinitely generated if \(M/U\) is finitely generated. If every cofinite submodule of \(M\) has an (ample) \(F\)-supplement, then \( M\) is an (amply) cofinitely \(F\)-supplemented module.\N\NThe author proves several properties of (amply) (cofinitely) \(F\)-supplemented modules, and characterises them in terms of their submodules and factor modules.