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Simple rings and degree maps
University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.ORCID iD: 0000-0001-6594-7041
Lund University, Centre for Mathematical Sciences.
2014 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 401, p. 201-219Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2014. Vol. 401, p. 201-219
Keywords [en]
Degree map, Ideal associativity, Ring extension, Simplicity
National Category
Algebra and Logic
Research subject
ENGINEERING, Mathematics
Identifiers
URN: urn:nbn:se:hv:diva-5902DOI: 10.1016/j.jalgebra.2013.11.023ISI: 000330599500011Scopus ID: 2-s2.0-84891812645OAI: oai:DiVA.org:hv-5902DiVA, id: diva2:692909
Available from: 2014-02-03 Created: 2014-02-03 Last updated: 2019-11-29Bibliographically approved

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