Epsilon-strongly graded rings, separability and semisimplicity

Abstract

We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring is separable over its principal component. Thereby, we simultaneously generalize a result for strongly group graded rings by Nǎstǎsescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the class of unital partial crossed products appear in the class of epsilon-strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings. Thereby, we obtain, in the special case of unital partial crossed products, a short proof of a general result by Dokuchaev, Exel and Simón concerning when graded rings can be presented as partial crossed products. We also provide some interesting classes of examples of separable epsilon-strongly graded rings, with finite as well as infinite grading groups. In particular, we obtain an answer to a question raised by Le Bruyn, Van den Bergh and Van Oystaeyen in 1988.

Induced quotient group gradings of epsilon-strongly graded rings

“…30]) that an element s ∈ S g is called epsilon-invertible if there exists some element t ∈ S g −1 such that st = ǫ g and ts = ǫ g −1 . Furthermore, recall (see [16,Def. 32]) that (S, {S g } g∈G ) is called an epsilon-crossed product if there is an epsilon-invertible element in S g for all g ∈ G. Let G-ǫCROSS denote the category of epsilon-crossed products.…”

“…This property is of particular importance for the theory of strongly group graded rings initiated by Dade (see Section 2). The class of epsilon-strongly G-graded rings was introduced by Nystedt,Öinert and Pinedo [16] as a generalization of unital strongly G-graded rings. A G-grading {S g } g∈G of S is epsilonstrong if any only if, for every g ∈ G, there is an element ǫ g ∈ S g S g −1 such that for all s ∈ S g the equations ǫ g s = s = sǫ g −1 hold (see [16,Prop.…”

“…a, b ∈ F =⇒ a ∨ b ∈ F provided that the upper bound a ∨ b exists. Recall (see [16,Prop. 5]) that if {S g } g∈G is epsilon-strong, then the elements ǫ g are central idempotents of the principal component S e .…”

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

“…Next, we recall two special types of group graded rings generalizing the classical notion of unital strongly group graded rings. Nystedt, Oinert and Pinedo [24] recently introduced the class of epsilon-strongly G-graded rings:…”

Bimodules in Group Graded Rings

“…crystalline graded [9] or epsilon-strongly graded [10], then it is necessarily strongly G-graded. This follows from Proposition 3 and the fact that both crystalline graded rings and epsilon-strongly graded rings are left (and right) nondegenerate.…”