Object-unital groupoid graded rings, crossed products and separability

Abstract

We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizing previous results from the unital case to a non-unital situation.

Chain conditions for rings with enough idempotents with applications to category graded rings

“…Recall from [6,7] that this means that there for every g ∈ G 1 is an additive subgroup S g of S such that S = g∈G 1 S g and for all g, h ∈ G 1 , the inclusion S g S h ⊆ S gh holds, if (g, h) ∈ G 2 , and S g S h = {0}, otherwise. Note that if H is a subcategory of G then S H := ⊕ h∈H 1 S h is a subring of S. Following [2,3] (see also [8]) we say that the G-grading on S is object unital if for all a ∈ G 0 the ring S a is unital and for all g ∈ G 1 and all s ∈ S g the equalities 1 S c(g) s = s1 S d(g) = s hold. In that case, S is a ring with enough idempotents with {1 Sa } a∈G 0 as a complete set of idempotents and the following equality holds for all a, b ∈ G 0 :…”

“…Then S is left/right artinian if and only if G 0 is finite and for every a ∈ G 0 the ring S a is left/right artinian and S G(a) is finitely generated as a left/right S a -module. (2) Let G be polycyclic-by-finite. Then S is left/right noetherian if and only if G 0 is finite and for every a ∈ G 0 the ring S a is left/right noetherian.…”