Matrix ring structure of some skew group rings

If S is a ring with 1, and G is a finite group acting faithfully as automorphisms of S, then it is well known that the skew group ring S ∗ G is a separable extension of S if and only if there exists a central element in S with trace one. A ring is an Azumaya algebra if it is separable over its center. The case of when the group ring S[G] is Azumaya was studied by De Meyer and Janusz in [2] ; and lately the case of twisted group rings was studied by Szeto and Wong in [5] ; but the techniques used on those cases cannot be applied to the skew group rings precisely because the elements of the ring S do not commute with the elements of the group G. The purpose of this paper is to give conditions on the action of the group in order to make the skew group ring an Azumaya algebra. A ring A is said to be separable over a subring B if the (A − A)-module homomorphism of A⊗BA onto A defined by a⊗b 7−→ ab splits, and A is called H-separable over B if A ⊗B A is isomorphic as (A − A)-bimodule to a direct summand of A for some n ≥ 1. Clearly H-separable extension are separable. Let S be a ring with 1 and G be a finite group acting as automorphisms of S with fixed ring S; that we will denote R. The skew group ring S ∗G is the free left S-module with basis G, where multiplication is defined according

Given a group G acting on a ring R via α:G → Aut (R), one can construct the skew group ring R*αG. Skew group rings have been studied in depth, but necessary and sufficient conditions for the simplicity of a general skew group ring are not known. In this paper, such conditions are given for certain types of skew group rings, with an emphasis on Von Neumann regular skew group rings. Next the results of the first section are used to construct a class of simple skew group rings. In particular, we obtain a more efficient proof of the simplicity of a certain ring constructed by J. Trlifaj.

We are interested in the McKay quiver Γ(G) and skew group rings A ∗G, where G is a finite subgroup of GL(V ), where V is a finite dimensional vector space over a field K, and A is a K −G-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups mathsf {G} subseteq text {GL}(V) and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups G(r,p,n). We first look at the case G(1,1,n), which is isomorphic to the symmetric group Sn, followed by G(r,1,n) for r > 1. Then, using Clifford theory, we can determine the McKay quiver for any G(r,p,n) and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra widetilde {A}(mathsf {G}) of a finite group mathsf {G} subseteq text {GL}(V), which is Morita equivalent to the skew group ring A ∗G. This description gives us an embedding of the basic algebra Morita equivalent to A ∗ G into a matrix algebra over A.

In this paper, we investigate the fixed submodules of modules over skew group rings. We give two applications, the first concerns the determination of a simple basis of the center of a skew group ring. In the second application, we characterize in terms of the structure groups all the mod-retractable skew group rings. More precisely, we prove that if [Formula: see text] is a group acting on a field [Formula: see text] with kernel [Formula: see text], then the skew group ring [Formula: see text] is mod-retractable if and only if [Formula: see text] is mod-retractable and [Formula: see text] is of finite index in [Formula: see text].

It is well known that the set of isomorphism classes of extensions of groups with abelian kernel is characterized by the second cohomology group. In this paper we generalise this characterization of extensions to a natural class of extensions of monoids, the cosetal extensions. An extension is cosetal if for all g,g' in G in which e(g) = e(g'), there exists a (not necessarily unique) n in N such that g = k(n)g'. These extensions generalise the notion of special Schreier extensions, which are themselves examples of Schreier extensions. Just as in the group case where a semidirect product could be associated to each extension with abelian kernel, we show that to each cosetal extension (with abelian group kernel), we can uniquely associate a weakly Schreier split extension. The characterization of weakly Schreier split extensions is combined with a suitable notion of a factor set to provide a cohomology group granting a full characterization of cosetal extensions, as well as supplying a Baer sum.

In this chapter, the theory of ring hulls is used to determine when various ring extensions are in the classes of interest (e.g., right (FI-) extending, (quasi-) Baer, etc.) or when certain subrings (e.g., the fixed ring) are in these classes. Section 9.1 begins with a characterization of a right extending ring whose maximal right ring of quotients is the 2×2 matrix ring over a division ring. This result eventually leads to a characterization of all right rings of quotients of a 2×2 upper triangular matrix ring over a commutative PID which are right extending, Baer, right Rickart, or right semihereditary. Skew group rings and fixed rings are considered in Sect. 9.2. The main results of this section concern semiprime rings with a group of X-outer ring automorphisms which have their skew group ring and/or fixed ring being quasi-Baer. In the final section, various matrix ring extensions (both finite and infinite) and monoid ring extensions of a ring hull are compared to the corresponding ring hull of the matrix or monoid ring extension. Moreover, for a semiprime ring R which is Morita equivalent to a ring S, then their quasi-Baer ring hulls are also Morita equivalent.

This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category 𝒪. We study functorial properties of 𝒪 across various setups. The first setup is over a skew group ring, involving a finite group Γ acting on a regular triangular algebra A. We develop Clifford theory for A⋊Γ, and obtain results on block decomposition, complete reducibility, and enough projectives. 𝒪 is shown to be a highest weight category when A satisfies one of the “Conditions (S);” the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module. Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories 𝒪; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup, and which yield information about 𝒪.

If [Formula: see text] is a directed graph and [Formula: see text] is a field, the Leavitt path algebra [Formula: see text] of [Formula: see text] over [Formula: see text] is naturally graded by the group of integers [Formula: see text] We formulate properties of the graph [Formula: see text] which are equivalent with [Formula: see text] being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of [Formula: see text] are also characterized in terms of the pre-ordered group properties of the Grothendieck [Formula: see text]-group of [Formula: see text]. If [Formula: see text] has finitely many vertices, we characterize when [Formula: see text] is strongly graded in terms of the properties of [Formula: see text] Our proof also provides an alternative to the known proof of the equivalence [Formula: see text] is strongly graded if and only if [Formula: see text] has no sinks for a finite graph [Formula: see text] We also show that, if unital, the algebra [Formula: see text] is strongly graded and graded unit-regular if and only if [Formula: see text] is a crossed product. In the process of showing the main result, we obtain conditions on a group [Formula: see text] and a [Formula: see text]-graded division ring [Formula: see text] equivalent with the requirements that a [Formula: see text]-graded matrix ring [Formula: see text] over [Formula: see text] is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group [Formula: see text] on the Grothendieck [Formula: see text]-group [Formula: see text]

A ring [Formula: see text] is called semiboolean if [Formula: see text] is boolean and idempotents lift modulo [Formula: see text], where [Formula: see text] denotes the Jacobson radical of [Formula: see text]. In this paper, we define [Formula: see text]-boolean rings as a generalization of semiboolean rings. A ring [Formula: see text] is said to be J-boolean if [Formula: see text] is boolean. Various basic properties of these rings are obtained. The [Formula: see text]-boolean group rings and skew group rings have been studied. It is investigated whether the results obtained for [Formula: see text]-boolean group rings also hold for the skew group rings.

Strongly graded rings and almost split sequences

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G T=<1>,where G T is the inertial group of R.

Homological Dimension of Skew Group Rings and Crossed Products

Necessary and sufficient conditions for simplicity of a general skew group ring A ⋊σ G are not known. In this article, we show that a skew group ring A ⋊σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. As an application, we show that a transformation group (X, G), where X is a compact Hausdorff space acted upon by an abelian group G, is minimal and faithful if and only if its associated skew group algebra C(X) ⋊σ G is simple.

Let R be a commutative ring, let G be a finite group acting on R as automorphisms of R and let R * G be the skew group ring. By using the decomposition subgroups of G, the inertial subgroups of G, the properties of the coefficient ring R and the properties of the fixed subring RG, some necessary and sufficient conditions for R * G to be a prime Goldie ring, a semi-hereditary order in a simple Artinian ring, or a Prufer order in a simple Artinian ring are given.

Let [Formula: see text] be a finite group acting on a ring [Formula: see text] and [Formula: see text] a subgroup of [Formula: see text]. In this paper, we compare some homological dimensions over the skew group rings [Formula: see text] and [Formula: see text]. Moreover, under the assumption that [Formula: see text] is a separable extension over [Formula: see text], we show that the skew group rings [Formula: see text] and [Formula: see text] share some properties such as being [Formula: see text]-Gorenstein, [Formula: see text]-perfect, [Formula: see text]-coherent, [Formula: see text], Ding–Chen or IF-rings.