A Generalization of Group-Graded Modules

Abstract

In this article, we generalize the concept of group-graded modules by introducing the concept of G-weak graded R-modules, which are R-modules graded by a set G of left coset representatives, where R is a G-weak graded ring. Moreover, we prove some properties of these modules. Finally, results related to G-weak graded fields and their vector spaces are deduced. Many considerable examples are provided with more emphasis on the symmetric group S3 and the dihedral group D6, which gives the group of symmetries of a regular hexagon.

Graded Rings Associated with Factorizable Finite Groups

“…In ref. [11], Beggs constructs an algebraic structure consisting of a set G of left coset representatives for the left action of a subgroup H on a group X and a binary operation ' * ' on G. This operation guarantees the left identity and the right division property on G. ' * ' is not associative in the standard way, though the associativity could be satisfied by applying a "cocycle" f : G × G −→ H. Based on this algebraic structure (G, * ) and the cocycle f together with the action : G × H → H and the coaction : G × H → G defined in [11], many research articles on the non-trivially associated categories and the G-weak graded rings and modules have been published (see [12][13][14][15]). The independence on the choice of representatives was proven in [11].…”

“…In refs. [13][14][15], the concepts of the group graded rings and modules were generalized by using the set G of left coset representatives, which was mentioned above with the binary operation ' * ' defined on it. The new generalized concepts were named G-weak graded rings and G-weak graded modules.…”

“…It was noted that the grading by a set of left coset representatives with the binary operation ' * ' is not always applicable; see [14,15]. In this article, we introduce two different approaches to make it applicable for any ring R with unity.…”