Cofinitely \(\delta_M\)-supplemented and cofinitely \(\delta_M\)-semiperfect modules. (Q2911219)

The notation \(\sigma[M]\) is used for the full subcategory of right \(R\)-modules whose objects are all right \(R\)-modules subgenerated by \(M\) (\(R\) an associative ring with unity and \(M\) a right \(R\)-module). A module \(N\) in \(\sigma[M]\) is defined to be cofinitely \(\delta_M\)-supplemented (resp. \(\oplus\)-cofinitely \(\delta_M\)-supplemented) provided each cofinite submodule of \(N\) has a \(\delta_M\)-supplement in \(N\) (resp. a \(\delta_M\)-supplement in \(N\) which is a direct summand of \(N\)). These concepts are generalizations of \(\delta_M\)- (resp. \(\oplus\delta_M\))-supplemented modules. Factor modules and direct sums of cofinitely \(\delta_M\)-supplemented modules are shown to retain this property. Cofinitely \(\delta_M\)-supplemented modules are characterized as those for which every maximal submodule of \(N\) has a \(\delta_M\)-supplement in \(N\).NEWLINENEWLINE A module \(N\) in \(\sigma[M]\) is said to be amply cofinitely \(\delta_M\)-supplemented if each cofinite submodule of \(N\) has an ample \(\delta_M\)-supplement in \(N\), i.e. if every submodule \(K\) of \(N\) with \(N=L+K\) contains a \(\delta_M\)-supplement \(L\) of \(N\). Modules in \(\sigma[M]\) all of whose submodules are cofinitely-\(\delta_M\)-supplemented, are shown to be amply cofinitely-\(\delta_M\)-supplemented, as are \(\pi\)-projective cofinitely-\(\delta_M\)-supplemented modules in \(\sigma[M]\). In the case of duo or distributive modules, factor modules are shown to retain the \(\oplus\)-cofinitely-\(\delta_M\)-supplemented property.NEWLINENEWLINE An epimorphism \(f\colon P\to N\), \(P,N\in\sigma[M]\), is defined to be a \(\delta_M\)-cover if \(\text{Ker}(f)\) is a \(\delta_M\)-small submodule of \(P\). If \(P\) is projective in \(\sigma[M]\), then \(f\) is called a projective \(\delta_M\)-cover. If a factor module by a cofinite submodule of a module \(K\) in \(\sigma[M]\) has a projective \(\delta_M\)-cover, then \(K\) is called a cofinitely \(\delta_M\)-semiperfect module. It is shown that cofinitely-\(\delta_M\)-semiperfect modules in \(\sigma[M]\) are cofinitely-\(\delta_M\)-supplemented. Homomorphic images of cofinitely-\(\delta_M\)-semiperfect modules are again cofinitely-\(\delta_M\)-semiperfect. The concepts of cofinitely-\(\delta_M\)-semiperfect, amply cofinitely \(\delta_M\)-supplemented and cofinitely-\(\delta_M\)-supplemented modules are shown to be equivalent under certain circumstances. A projective module in \(\sigma[M]\) is shown to be cofinitely \(\delta_M\)-semiperfect if and only if \(N\) is \(\oplus\)-cofinitely \(\delta_M\)-supplemented and an arbitrary direct sum of projective modules in \(\sigma[M]\) is shown to be cofinitely \(\delta_M\)-semiperfect if and only if each summand is. More characterizations of projective modules in \(\sigma[M]\) to be cofinitely \(\delta_M\)-semiperfect conclude the paper.