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- Nystedt, Patrik, 1970-, et al.
(författare)
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Simple rings and degree maps
- 2014
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Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 401, s. 201-219
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Tidskriftsartikel (refereegranskat)abstract
- 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 A I ⊆ I A. 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-zero a ' ∈ I with d(a ') ≤ d(a) and d(a 'b - ba ') < d(a) for all b ∈ X. We 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, A C A = 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. © 2013 Elsevier Inc.
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- Nystedt, Patrik, 1970-, et al.
(författare)
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Simple semigroup graded rings
- 2015
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Ingår i: Journal of Algebra and its Applications. - 0219-4988 .- 1793-6829. ; 14:7
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Tidskriftsartikel (refereegranskat)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 nonzero elements of eGe form a hypercentral group and R-e has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring fR(eGe)f is a field. This is a generalization of a result of 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 Goncalves’. We also point out how Jespers’ result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.
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