Rad-Supplemented Modules

Abstract

-Let t be a radical for the category of left R-modules for a ring R. If M is a t-coatomic module, that is, if M has no nonzero t-torsion factor module,

ON ω-LOCAL MODULES AND Rad-SUPPLEMENTED MODULES

“…It is clear that R (R/I) is Rad-supplemented if and only if (R/I) (R/I) is Radsupplemented. So without loss of generality we can assume that I = 0 and M = R. Since R R is Rad-supplemented, R/P (R) is semiperfect by [4,Theorem 6.5]. Let J be the Jacobson radical of R. Then the ring R/J ∼ = (R/P (R))/(J/P (R)) is semisimple.…”

Rad-Supplementing Modules

“…Hence F is ample Rad-supplementing by Proposition 3.1. Conversely, if F is ample Rad-supplementing, then it is Rad-supplemented by Corollary 3.2, and so R is left perfect by [6,Theorem 5.3].…”

“…M is called a τ -supplemented module if every submodule of M has a τ -supplement in M . For the particular case τ = Rad, Rad-supplemented modules have been studied in [6]; rings over which all modules are Rad-supplemented were characterized. Also, in the recent paper [7], the relation between Rad-supplemented modules and local modules have been investigated.…”

“…In the proof of this result, Rad-supplemented modules plays an important role as, of course, every ample Rad-supplementing module is Rad-supplemented. Finally, using the characterization of Rad-supplemented modules given in [6], we characterize the rings over which every module is (ample) Rad-supplementing. We prove that every left R-module is (ample) Radsupplementing if and only if every reduced left R-module is Rad-supplementing if and only if R/P (R) is left perfect; Theorem 3.4.…”

Rad-⊕-Supplemented Modules

“…100]. Adapting the concept of supplemented modules, we say that M is Rad-supplemented if every submodule has a Rad-supplement in M , and M is Rad-⊕-supplemented if every submodule has a Rad-supplement that is a direct summand of M [4,7]. Under given definitions, we clearly have the following implication on modules:…”