On a Recent Generalization of Semiperfect Rings

Abstract

In a recent paper by Wang and Ding, it was stated that any ring which is generalized supplemented as a left module over itself is semiperfect. The purpose of this note is to show that Wang and Ding's claim is not true and that the class of generalized supplemented rings lies properly between the classes of semilocal and semiperfect rings. Moreover, we propose a corrected version of the theorem by introducing a wider notion of 'local' for submodules.2000 Mathematics subject classification: primary 16L30; secondary 16D10.

ON ω-LOCAL MODULES AND Rad-SUPPLEMENTED MODULES

“…By (ii), there exists a submodule E of L such that E is a Rad-supplement of K in M . By [3,Lemma 3.3], E is a w-local submodule of L. Note that U = U + E, since otherwise we have E ⊆ U ⊆ K and K = K + E = M . It follows that M = U + E + F for some element F ∈ S. But E + F ∈ S, a contradiction.…”

“…w-local modules were first studied by Ware in [18], Gerasimov and Sakhaev in [8]. In [3], Büyükaşik and Lomp showed that this type of modules play a key role in the study of Rad-supplemented modules. This role is as important as the role played by local modules for supplemented modules.…”

“…This role is as important as the role played by local modules for supplemented modules. In fact, it is shown (in [3]) that any Rad-supplement of a maximal submodule is w-local and any finitely generated Rad-supplemented module is a sum of finitely many w-local submodules. But these modules may have a complicated structure.…”

tg-Supplemented Modules

“…Let M be an R-module and U , V be submodules of M . V is called a generalized supplement ( [1], [9], [11]…”

Generalized Cofinitely δ-Semiperfect Modules

“…Clearly supplemented modules are cofinitely supplemented and weakly supplemented modules are cofinitely weak supplemented. M is called cofinitely generalized supplemented if every cofinite submodule of M has a generalized supplement (see [3]). Since every submodule of a finitely generated module is cofinite, a finitely generated module is generalized supplemented if and only if it is cofinitely generalized supplemented.…”