Crossed Product-Like and Pre-Crystalline Graded Rings

Öinert, Johan; Silvestrov, Sergei

doi:10.1007/978-3-540-85332-9_24

Cited by 8 publications

(7 citation statements)

References 5 publications

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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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Section: Introductionmentioning

confidence: 99%

Maximal Commutative Subrings and Simplicity of Ore Extensions

2013

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“…R ′ coincides with its commutant in R. A lot of work has been devoted to the investigation of the connection between maximal commutativity and the ideal intersection property (see [2], [5], [6], [9], [10], [11], [15] and [25]). Recently (see [20], [21], [22], [23] and [24]) such a connection was established for the commutant of the neutral component of strongly group graded rings and crystalline graded rings (see Theorem 1 and Theorem 2 below).…”

Section: Introductionmentioning

confidence: 95%

The Ideal Intersection Property for Groupoid Graded Rings

Öinert

Lundström

2012

Communications in Algebra

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Abstract. We show that if a groupoid graded ring has a grading satisfying a certain nondegeneracy property, then the commutant of the center of the principal component of the ring has the ideal intersection property, that is it intersects nontrivially every nonzero ideal of the ring. Furthermore, we show that for skew groupoid algebras with commutative principal component, the principal component is maximal commutative if and only if it has the ideal intersection property.

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“…For related results concerning the ideal structure in skew group algebras and, more generally, group graded rings and Ore extensions, see [22], [23] [24], [25], [26], [33], [34] and [35].…”

Section: Introductionmentioning

confidence: 99%

Simple Skew Category Algebras Associated with Minimal Partially Defined Dynamical Systems

Lundström¹,

Öinert²

2012

Preprint

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In this article, we continue our study of category dynamical systems, that is functors s from a category G to Top op , and their corresponding skew category algebras. Suppose that the spaces s(e), for e ∈ ob(G), are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) G is inverse connected, (iii) s is minimal and (iv) s is faithful. We also show that if G is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by Öinert for skew group algebras to a large class of skew category algebras.

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Crossed Product-Like and Pre-Crystalline Graded Rings

Cited by 8 publications

References 5 publications

Maximal Commutative Subrings and Simplicity of Ore Extensions

Maximal Commutative Subrings and Simplicity of Ore Extensions

The Ideal Intersection Property for Groupoid Graded Rings

Simple Skew Category Algebras Associated with Minimal Partially Defined Dynamical Systems

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