Induced quotient group gradings of epsilon-strongly graded rings

Abstract

Let G be a group and let S = g∈G Sg be a G-graded ring. Given a normal subgroup N of G, there is a naturally induced G/N -grading of S. It is well-known that if S is strongly G-graded, then the induced G/N -grading is strong for any N . The class of epsilon-strongly graded rings was recently introduced by Nystedt,Öinert and Pinedo as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilonstrong. Moreover, we give necessary and sufficient conditions for the induced G/N -grading of an epsilon-strongly G-graded ring to be epsilon-strong. Our method involves relating different types of rings equipped with local units (s-unital rings, rings with sets of local units, rings with enough idempotents) with generalized epsilon-strongly graded rings.

Prime group graded rings with applications to partial crossed products and Leavitt path algebras

“…For every C ∈ G/N , put S C := x∈C S x . This yields a G/N -grading on S. The following non-trivial result, proved by Lännström, will be essential later on in this article: Proposition 2.20 ([22,Prop. 5.8]).…”

“…Now, we recall two important functorial constructions. For more details, we refer the reader to [22]. The first construction assigns a subring of S with an inherited grading.…”

The graded structure of algebraic Cuntz-Pimsner rings

“…However, in this article we will only consider the special case of Z-graded rings. (b) If S is epsilon-strongly Z-graded, then S is a unital ring (see [14,Prop. 3.8]).…”

“…(cf. [14,Expl. 4.5]) Let K be a unital ring and consider the infinite discrete graph E consisting of countably infinitely many vertices but no edges.…”

Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

“…This proves that unital strongly G-graded rings are epsilon-strongly G-graded. (b) If S is an epsilon-strongly G-graded ring, then S is a unital ring (see [19,Prop. 3.8]).…”