The Ideal Intersection Property for Groupoid Graded Rings

Öinert, Johan; Lundström, Patrik

doi:10.1080/00927872.2011.559181

Cited by 22 publications

(16 citation statements)

References 25 publications

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“…Recall also that the centralizer of a nonempty subset S of a ring R, which we denote by C R (S), is the set of all elements of R that commute with each element of S. If C R (S) = S holds, then S is said to be a maximal commutative subring of R. Note that a maximal commutative subring is necessarily commutative. Following [12], a subring S of a ring R is said to have the ideal intersection property in R, if S ∩ I ≠ {0} holds for each non-zero ideal I of R.…”

Section: Introductionmentioning

confidence: 99%

Simplicity of Partial Skew Group Rings of Abelian Groups

Gonçalves

2014

Can. math. bull.

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Abstract. Let A be a ring with local units, E a set of local units for A, G an abelian group and α a partial action of G by ideals of A that contain local units. We show that A⋆ α G is simple if and only if A is G-simple and the center of the corner eδ 0 (A⋆ α G)eδ 0 is a field for all e ∈ E. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

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Section: Introductionmentioning

confidence: 99%

Simplicity of Partial Skew Group Rings of Abelian Groups

Gonçalves

2014

Can. math. bull.

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“…These questions have been considered recently for algebraic crossed products and Banach algebra crossed products, both in the traditional context of crossed products by groups as well as generalizations to graded rings, crossed products by groupoids and general categories in [15,20,26,27,28,29,30,31,32,33,37,38,39,40].…”

Section: Introductionmentioning

confidence: 99%

Maximal Commutative Subrings and Simplicity of Ore Extensions

2013

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Communicated by T. LenaganThe aim of this paper is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x; id R , δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x; σ, δ] is a maximal commutative subring containing R and, in the case when σ = id R , we show that it intersects every nonzero ideal of R[x; id R , δ] nontrivially. Using this we show that if R is δ-simple and maximal commutative in R[x; id R , δ], then R[x; id R , δ] is simple. We also show that under some conditions on R the converse holds.

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“…The ideal intersection property has been studied in e.g. [18]. In this paper, subalgebras having this property are said to be essential and they are defined as follows.…”

Section: Essential Subalgebras Of Tgwasmentioning

confidence: 99%

“…Graded rings with this property have been studied before [4]. Then we can apply the general result from [18,Theorem 3]. We introduce the notion of regularly graded algebras, which is a generalization of crystalline graded algebras, and provide exact conditions for a TGWA to be regularly graded (Theorem 4.3).…”

Section: Introductionmentioning

confidence: 99%

Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

Hartwig

Öinert

2013

Journal of Algebra

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In this paper we show that each non-zero ideal of a twisted generalized Weyl algebra (TGWA) A intersects the centralizer of the distinguished subalgebra R in A non-trivially. We also provide a necessary and sufficient condition for the centralizer of R in A to be commutative, and give examples of TGWAs associated to symmetric Cartan matrices satisfying this condition. By imposing a certain finiteness condition on R (weaker than Noetherianity) we are able to make an Ore localization which turns out to be useful when investigating simplicity of the TGWA. Under this mild assumption we obtain necessary and sufficient conditions for the simplicity of TGWAs. We describe how this is related to maximal commutativity of R in A and the (non-) existence of non-trivial Z n -invariant ideals of R. Our result is a generalization of the rank one case, obtained by D. A. Jordan in 1993. We illustrate our theorems by considering some special classes of TGWAs and providing concrete examples.

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The Ideal Intersection Property for Groupoid Graded Rings

Cited by 22 publications

References 25 publications

Simplicity of Partial Skew Group Rings of Abelian Groups

Simplicity of Partial Skew Group Rings of Abelian Groups

Maximal Commutative Subrings and Simplicity of Ore Extensions

Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

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