On graded weak multiplication modules

Abstract

Abstract. Let G be a group with identity e, and let R be a G-graded commutative ring, and let M be a graded R-module. In this paper we characterize graded weak multiplication modules.

Generalizations of graded second submodules

“…Indeed, S −1 R = g∈G (S −1 R) g where (S −1 R) g = {r/s : r ∈ R, s ∈ S and g = (degs) −1 (degr)}. We write h(S −1 R) = g∈G (S −1 R) g [8]. Proof.…”

“…A proper graded submodule N of a graded R-module M is said to be graded prime, if r g m h ∈ N where r g ∈ h(R) and m h ∈ h(M), then m h ∈ N or r g ∈ (N : M). A graded R-module M is called graded prime, if the zero graded submodule is graded prime in M. For more information about graded prime submodules over commutative graded rings see [3,7,9]…”

Graded S-1-absorbing prime submodules in graded multiplication modules

“…Gradings appear in many circumstances, both in elementary and advanced level. In recent years, rings with a group-graded structure have become increasingly important and consequently, the graded analogues of different concepts are widely studied (see [1], [4], [7], [8], [9], [10], [11], [12], [13] and [14]). In the third section of this paper, we introduce and study the notion of graded S-1-absorbing prime submodules of a graded R-module M as a generalization of graded prime submodules and we investigate some properties of such graded submodules.…”

On graded $S$-comultiplication modules

“…Recently, extensive researches have been done on rings with groupgraded structure, see for example [2,3,5,13,17,20]. The notion of graded multiplication modules was studied by many authors, see for example [7,9,12,21]. The notion of graded comultiplication modules which are the dual nation of graded multiplication modules was introduced and studied by Ansari-Toroghy and Farshadifar in [2].…”