Simple group graded rings and maximal commutativity

Öinert, Johan

doi:10.1090/conm/503/09899

Cited by 17 publications

(28 citation statements)

References 10 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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“…Now suppose that s is both minimal and faithful. We show that A i is simple for i = 1 (the case i = 1 was already treated in [8,13]). Again faithfulness is equivalent to injectivity of σ.…”

Section: Topological Dynamicsmentioning

confidence: 76%

“…In particular, Power [14] has shown that if X is a compact Hausdorff space of infinite cardinality, then C * (X, s) is simple if and only if (X, s) is minimal. Inspired by this result, Öinert [8,13] has shown that there is an analogous result for skew group algebras defined by general group topological dynamical systems. In fact Öinert loc.…”

Section: Introductionmentioning

confidence: 79%

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Simple rings and degree maps

Nystedt

Öinert

2014

Journal of Algebra

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For an extension A/B of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of A with a property that we call A-simplicity of B. By this we mean that there is no non-trivial ideal I of B being A-invariant, that is satisfying AI ⊆ IA. We show that A-simplicity of B is a necessary condition for simplicity of A for a large class of ring extensions when B is a direct summand of A. To obtain sufficient conditions for simplicity of A, we introduce the concept of a degree map for A/B. By this we mean a map d from A to the set of non-negative integers satisfying the following two conditions (d1) if a ∈ A, then d(a) = 0 if and only if a = 0; (d2) there is a subset X of B generating B as a ring such that for each non-zero ideal I of A and each non-zero a ∈ I there is a non-zeroWe show that if the centralizer C of B in A is an A-simple ring, every intersection of C with an ideal of A is A-invariant, ACA = A and there is a degree map for A/B, then A is simple. We apply these results to various types of graded and filtered rings, such as skew group rings, Ore extensions and Cayley-Dickson doublings.

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“…R e is simple (see Proposition 5); 2. R e is commutative (see [11,Theorem 6.6]); 3. G is a hypercentral group (see Proposition 6).…”

Section: Open Questionsmentioning

confidence: 99%

Bimodules in Group Graded Rings

Öinert

2017

Algebr Represent Theor

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In this article we introduce the notion of a controlled group graded ring. Let G be a group, with identity element e, and let R = ⊕ g∈G R g be a unital G-graded ring. We say that R is G-controlled if there is a one-to-one correspondence between subsets of the group G and (mutually non-isomorphic) R e -sub-bimodules of R, given by G ⊇ H → ⊕ h∈H R h . For strongly G-graded rings, the property of being G-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general G-graded ring to be G-controlled. We also give a characterization of strongly G-graded rings which are G-controlled. As an application of our main results we give a description of all intermediate subrings T with R e ⊆ T ⊆ R of a G-controlled strongly G-graded ring R. Our results generalize results for artinian skew group rings which were shown by Azumaya 70 years ago. In the special case of skew group rings we obtain an algebraic analogue of a recent result by Cameron and Smith on bimodules in crossed products of von Neumann algebras.

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Simple group graded rings and maximal commutativity

Cited by 17 publications

References 10 publications

Maximal Commutative Subrings and Simplicity of Ore Extensions

Maximal Commutative Subrings and Simplicity of Ore Extensions

Simple rings and degree maps

Bimodules in Group Graded Rings

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