On a new variation of injective modules

Abstract

In this paper, we provide various properties of GE and GEEmodules, a new variation of injective modules. We call M a GE-module if it has a g-supplement in every extension N and, we call also M a GEE-module if it has ample g-supplements in every extension N. In particular, we prove that every semisimple module is a GE-module. We show that a module M is a GEE-module if and only if every submodule is a GE-module. We study the structure of GE and GEE-modules over Dedekind domains. Over Dedekind domains the class of GE-modules lies between W S-coinjective modules and Zöschinger's modules with the property (E). We also prove that, if a ring R is a local Dedekind domain, an R-module M is a GE-module if and only if M = (R) n K N , where R is the completion of R, K is injective and N is a bounded module.

MODULES WITH THE PROPERTY Radg

“…2 and Lemma 1 in[1]. If X is a semisimple module, then X has the property Rad g .The next example shows that modules with the property Rad g are a proper generalization of the class of  …”

“…Using Theorem 21 [1], we can easily obtain the following proposition. Proof: Suppose that K is any submodule of X .…”

“…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.…”