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Non-associative Ore extensions
University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.ORCID iD: 0000-0001-6594-7041
Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, Sweden.
Mälardalen University, Academy of Education, Culture and Communication,Västerås, Sweden.
2018 (English)In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 224, no 1, p. 263-292Article in journal (Refereed) Published
Abstract [en]

We introduce non-associative Ore extensions, S = R[X; sigma, delta], for any non-ssociative unital ring R and any additive maps sigma, delta : R -> R satisfying sigma(1) = 1 and delta(1) = 0. In the special case when delta is either left or right R-delta-linear, where R-delta = ker(delta), and R is delta-simple, i.e. 0 and R are the only delta-invariant ideals of R, we determine the ideal structure of the non-associative differential polynomial ring D = R[X; id(R),delta]. Namely, in that case, we show that all non-zero ideals of D are generated by monic polynomials in the center Z(D) of D. We also show that Z(D) = R-delta[p] for a monic p is an element of R-delta [X], unique up to addition of elements from Z(R)(delta) . Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of D to show that D is simple if and only if R is delta-simple and Z(D) equals the field R-delta boolean AND Z(R). This provides us with a non-associative generalization of a result by Oinert, Richter and Silve-strov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of D in the cases when the characteristic of the field R-delta boolean AND Z(R) is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.

Place, publisher, year, edition, pages
2018. Vol. 224, no 1, p. 263-292
National Category
Algebra and Logic
Research subject
ENGINEERING, Mathematics
Identifiers
URN: urn:nbn:se:hv:diva-12481DOI: 10.1007/s11856-018-1642-zISI: 000431796000010Scopus ID: 2-s2.0-85044256972OAI: oai:DiVA.org:hv-12481DiVA, id: diva2:1219181
Note

First Online: 06 March 2018

Available from: 2018-06-15 Created: 2018-06-15 Last updated: 2019-10-24Bibliographically approved

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