Applications of epigroups to graded ring theory

Kelarev, AV

doi:10.1007/bf02573530

Cited by 18 publications

(9 citation statements)

References 32 publications

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“…geG The following general problem was posed by E. Zelmanov (see [8]): find all G-gradings of the matrix algebra M n (k), where G is a group, k a field, and n a positive integer. The answer depends on the structure of G and A;, so it is hard to expect the problem can be solved in the general.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Group gradings of M₂(K)

Khazal¹,

Boboc²,

Dăscălescu³

2003

Bull. Austral. Math. Soc.

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We describe all group gradings of the matrix algebra M 2 (k), where k is an arbitrary field. We prove that any such grading reduces to a grading of type C 2 , a grading of type C 2 xC 2 , or to a good grading. We give new simple proofs for the description of C 2 -gradings and C 2 x C2-gradings on M 2 (k). INTRODUCTION AND PRELIMINARY RESULTSLet k be a field, G a group and A a /c-algebra. We say that A is G-graded if A = 0 A g , a direct sum of fc-vector subspaces, such that A g A h C A gh for any g,heG. geGThe following general problem was posed by E. Zelmanov (see [8]): find all G-gradings of the matrix algebra M n (k), where G is a group, k a field, and n a positive integer. The answer depends on the structure of G and A;, so it is hard to expect the problem can be solved in the general. However, several results have been obtained in special cases. In [7], gradings on A = M n (k) for which every matrix unit e^ (the matrix having 1 on the (i, j)-position, and zero elsewhere) is a homogeneous element (that is, it belongs to one of the subspaces A g ) have been studied. These gradings have been further investigated in [6], where they were called good gradings. Good gradings are fundamental in the study of all gradings, since as we shall see below, in certain cases any grading is isomorphic to a good grading. The description of all gradings of M 2 (k) by the cyclic group C 2 with two elements was done in [6], by using computational methods and the duality between group actions and group gradings. This was explained in [4] in terms of actions and coactions of Hopf algebras, the basic underlying idea being that a G-grading on an algebra A is precisely a structure of a ftG-comodule algebra on A. This idea was very useful for studying gradings of matrix algebras by cyclic groups, see [4]. In particular all the isomorphism types of C 2 -gradings on M 2 {k) were obtained in [4] in the case where char(fc) ^ 2. For char(fe) = 2, the classification has been completed in [2]. We note that in the case where k is algebraically closed, any C m -grading on M n (k) is isomorphic to a good grading (see [5,10]). This fact led to a different approach to the classification of all gradings of M n (k) over cyclic groups for an arbitrary field k, by using descent theory;

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Group gradings of M₂(K)

Khazal¹,

Boboc²,

Dăscălescu³

2003

Bull. Austral. Math. Soc.

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“…Group graded rings as well as Clifford theory for group graded rings were studied and their properties were investigated by many mathematicians, see for exampl [8], [9], [10], [12], [16], [21] and [22]. Nevertheless, rings and modules can be graded by using semigroups instead of groups leading to more general results as we can see in [1], [11], [13], [14], [15] and [19].…”

Section: Introductionmentioning

confidence: 99%

On Weak Graded Rings II

Al-Subaie

Al-Shomrani

2019

Eur. J. Pure Appl. Math.

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The G-weak graded rings are rings graded by a set G of left coset representatives for the left action of a subgroup H of a finite group X. The main aim of this article is to study the concept of G-weak graded rings and continue the investigation of their properties. Moreover, some results concerning G-weak graded rings of fractions are derived. Finally, some additional examples of G-weak graded rings are provided.

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“…An important problem in the investigation of graded rings is to explore how properties of R are connected to properties of subrings of R. In the associative case, many results of this sort are known for finiteness conditions, nil and radical properties, semisimplicity, semiprimeness and semiprimitivity (see e.g. [7,8]).…”

Section: Introductionmentioning

confidence: 99%

Simple graded rings, nonassociative crossed products and Cayley–Dickson doublings

Nystedt

Öinert

2019

J. Algebra Appl.

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We show that if a non-associative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers to a non-associative setting. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply our result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.

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Applications of epigroups to graded ring theory

Cited by 18 publications

References 32 publications

Group gradings of M₂(K)

Group gradings of M₂(K)

On Weak Graded Rings II

Simple graded rings, nonassociative crossed products and Cayley–Dickson doublings

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Applications of epigroups to graded ring theory

Cited by 18 publications

References 32 publications

Group gradings of M2(K)

Group gradings of M2(K)

On Weak Graded Rings II

Simple graded rings, nonassociative crossed products and Cayley–Dickson doublings

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Product

Resources

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Group gradings of M₂(K)

Group gradings of M₂(K)