Semigroup-graded rings with finite support

Clase, M. V.; Jespers, Eric; Río, Ángel del

doi:10.1017/s0017089500031190

Cited by 4 publications

(3 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the lemma below we use the idea of [9,Section 4] and show, in particular, that all semigroup regradings of elementary group gradings on M n (F ) can be reduced to group regradings.…”

Section: Regrading Full Matrix Algebras By Finite Groupsmentioning

confidence: 99%

On weak equivalences of gradings

Gordienko

Schnabel

2018

Journal of Algebra

View full text Add to dashboard Cite

When one studies the structure (e.g. graded ideals, graded subspaces, radicals,. . . ) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. The following question arises naturally: when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group? It turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading. Namely, a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group. The same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset. We show that if n 349, then on the full matrix algebra M n (F ) there exists an elementary group grading that is not weakly equivalent to any grading by a finite (semi)group, and if n 3, then any elementary grading on M n (F ) is weakly equivalent to an elementary grading by a finite group.be two gradings where S and T are (semi)groups and A and B are algebras.The most restrictive case is when we require that both grading (semi)groups coincide: Definition 1.1 (e.g. [11, Definition 1.15]). The gradings (1.1) are isomorphic if S = T and there exists an isomorphism ϕ : A → B of algebras such that ϕ(A (s) ) = B (s) for all s ∈ S. In this case we say that A and B are graded isomorphic. In some cases, such as in [16], less restrictive requirements are more suitable. Definition 1.2 ([16, Definition 2.3]). The gradings (1.1) are equivalent if there exists an isomorphism ϕ : A → B of algebras and an isomorphism ψ : S → T of (semi)groups such that ϕ(A (s) ) = B ψ(s) for all s ∈ S. Remark 1.3. The notion of graded equivalence was considered by Yu. A. Bahturin, S. K. Seghal, and M. V. Zaicev in [6, Remark after Definition 3]. In the paper of V. Mazorchuk and K. Zhao [19] it appears under the name of graded isomorphism. A. Elduque

show abstract

“…In the lemma below we use the idea of [9,Section 4] and show, in particular, that all semigroup regradings of elementary group gradings on M n (F ) can be reduced to group regradings.…”

Section: Regrading Full Matrix Algebras By Finite Groupsmentioning

confidence: 99%

On weak equivalences of gradings

Gordienko

Schnabel

2018

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…Thus, if it is possible, one can try to replace the grading group by a better one. For example, in [6] and [12] the authors studied the question, whether it is always possible to regrade a grading by a finite group. The situation, when this is possible, is very convenient since the algebra graded by a finite group G is an F G-comodule algebra and, in turn, an (F G) * -module algebra where F G is the group algebra of G, which is a Hopf algebra, and (F G) * is its dual.…”

Section: Introductionmentioning

confidence: 99%

Categories and Weak Equivalences of Graded Algebras

Gordienko

Schnabel

2019

Algebra Colloq.

View full text Add to dashboard Cite

When one studies the structure of graded algebras (e.g. graded ideals, graded subspaces, radicals,. . . ) or graded polynomial identities, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. Each group grading on an algebra can be weakly equivalent to G-gradings for many different groups G, however it turns out that there is one distinguished group among them called the universal group of the grading. In this paper we study categories and functors related to the notion of weak equivalence of gradings. In particular, we introduce an oplax 2-functor that assigns to each grading its support and show that the universal grading group functor has neither left nor right adjoint.

show abstract

“…The investigation of semigroup graded rings has been carried out by many authors, see e.g. [1,6,7,8,9,11,17,20,23,27,30,34,35,37]. For an excellent overview of the theory of semigroup graded rings, we refer the reader to A. V. Kelarev's extensive book [26], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Simple semigroup graded rings

Nystedt

Öinert

2015

J. Algebra Appl.

View full text Add to dashboard Cite

Abstract. We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the non-zero elements of eGe form a hypercentral group and R e has a non-zero idempotent f , then R is simple if and only if it is graded simple and the center of the corner subring f R eGe f is a field. This is a generalization of a result of E. Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of D. Gonçalves'. We also point out how E. Jespers' result immediately implies a generalization of a simplicity result, recently obtained by A. Baraviera, W. Cortes and M. Soares, for crossed products by twisted partial actions.

show abstract

Semigroup-graded rings with finite support

Cited by 4 publications

References 11 publications

On weak equivalences of gradings

On weak equivalences of gradings

Categories and Weak Equivalences of Graded Algebras

Simple semigroup graded rings

Contact Info

Product

Resources

About