On Weak Graded Rings II

Abstract

The G-weak graded rings are rings graded by a set G of left coset representatives for the left action of a subgroup H of a finite group X. The main aim of this article is to study the concept of G-weak graded rings and continue the investigation of their properties. Moreover, some results concerning G-weak graded rings of fractions are derived. Finally, some additional examples of G-weak graded rings are provided.

Graded Rings Associated with Factorizable Finite Groups

“…It is known, by definition, that in general R = s∈G R s is not necessarily a non-trivial G-weak graded ring; see for example [3,13]. However, in this section we define a relation between the operation * and the coaction that makes the ring R into a G-weak graded ring.…”

“…We recall that a subring K of a G-weak graded ring R is a G-weak graded subring if K itself is a G-weak graded ring [14]. In the next theorem, we discuss when a subring K of a G-weak graded ring R is a G-weak graded subring respecting the inclusion property of R that was mentioned in Theorem 2.…”

“…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.…”