Graded Rings Associated with Factorizable Finite Groups

Abstract

Let R be an associative ring with unity, X be a finite group, H be a subgroup of X, and G be a set of left coset representatives for the left action of H on X. In this article, we introduce two different ways to put R into a non-trivial G-weak graded ring that is a ring graded by the set G which is defined with a binary operation ∗ and satisfying an algebraic structure with specific properties. The first one is by choosing a subset S of G such that S is a group under the ∗ operation and putting Rt=0 for all t∈G and t∉S. The second way, which is the most important, is induced by combining the operation ∗ defined on G and the coaction ◁ of H on G. Many examples are provided.