Cofinitely Weak Supplemented Modules

Abstract

We prove that a module M is cofinitely weak supplemented or briefly cws (i.e., every submodule N of M with M=N finitely generated, has a weak supplement) if and only if every maximal submodule has a weak supplement. If M is a cws-module then every M-generated module is a cws-module. Every module is cws if and only if the ring is semilocal. We study also modules, whose finitely generated submodules have weak supplements.

Co-Coatomically Supplemented Modules

“…Afterwards ⊕-cofinitely supplemented, cofinitely semiperfect and cofinitely weak supplemented modules were studied intensively in the last ten years (see, e.g. (Çalışıcı and Pancar, 2004), (Çalışıcı and Pancar, 2005) and (Alizade and Büyükaşık, 2003)). …”

“…A module M is called cofinitely weak supplemented (cws-module in short) if every cofinite submodule has a weak supplement in M (Alizade and Büyükaşık, 2003).…”

“…For a module M, let Γ denote the set of all submodules K such that K is a weak supplement for some maximal submodule of M and cws(M) the sum of all submodules from Γ. cws(M) = 0 if Γ = ∅ (see (Alizade and Büyükaşık, 2003) …”

“…A module M is called ⊕-supplemented if every submodule has a supplement that is a direct summand of M. ⊕-supplemented modules are widely investigated in . If every maximal submodule has a weak supplement (see (Alizade and Büyükaşık, 2003)). …”

Generalized Cofinitely δ-Semiperfect Modules

“…M is called a cofinitely (weak) supplemented module if every cofinite submodule of M has a (weak) supplement in M (see [1], [2]). Clearly supplemented modules are cofinitely supplemented and weakly supplemented modules are cofinitely weak supplemented.…”