Rad-Supplementing Modules

Abstract

Abstract. Let R be a ring, and let M be a left R-module. If M is Radsupplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M . Any finite direct sum of Rad-supplementing modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal, reduced and the free left R-module ( R R) (N) is Rad-supplementing if and only if R is reduced and the free left R-module ( R R) (N) is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P (R) is left perfect, where P (R) is the sum of all left ideals I of R such that Rad I = I.

MODULES WITH THE PROPERTY Radg

“…It follows from Proposition 2.18 that K is Radsupplementing. Thus X is ample Radsupplementing by Proposition 3.1 [2].…”

“…In [10], Zöschinger characterized two properties in module categories as the property (E) and the property (EE), and it was determined the modules with the property (E) over nonlocal Dedekind ring R. Then it was defined (amply) Rad-supplementing modules as a proper generalization of property ((EE)) (E) in [2]. Based on these two studies, we examined the important algebraic properties that the module with the property (ample) Rad g provide by deriving from the notion gsupplement in [7].…”

“…Another variant of them are examined in [12]. In addition, (ample) Rad-supplementing are defined in [2]. On the other hand, another adaptation of these modules introduced in [1] and the authors named these modules as () GE  module.…”

“…In this article, modules with the property Rad g are defined as a new generalization of () GE  modules which was studied in [1] and some simple features of them are investigated. While preparing this article in addition to the article [2], the articles [3][4] connected with modules possessing a supplement and a Rad-supplement in each cofinite extension are examined. During the article, R will be an associative ring with identity and all modules are unital left Rmodules unless otherwise specified.…”

Modules that have a weak supplement in every extension

“…Let R be the product of the family fF i g i 2I , where each F i is a field for an infinite index set I . The ring R is a commutative Von Neumann regular but not hereditary [10,Example 2.15]. Then by [14, Section 23.5], R is a left V -ring.…”