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Non-associative Ore extensions
Högskolan Väst, Sweden.
Blekinge Tekniska Högskola, Sweden.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0003-3931-7358
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; σ, δ], for any nonassociative unital ring R and any additive maps σ, δ: R → R satisfying σ(1) = 1 and δ(1) = 0. In the special case when δ is either left or right Rδ-linear, where Rδ = ker(δ), and R is δ-simple, i.e. {0} and R are the only δ-invariant ideals of R, we determine the ideal structure of the nonassociative differential polynomial ring D = R[X; idR, δ]. 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) = Rδ[p] for a monic p ∈ Rδ[X], unique up to addition of elements from Z(R)δ. 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 δ-simple and Z(D) equals the field Rδ ∩ Z(R). This provides us with a non-associative generalization of a result by Öinert, Richter and Silvestrov. 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 Rδ ∩ 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
Keywords [en]
Ore extensions, Simplicity
National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-40323DOI: 10.1007/s11856-018-1642-zISI: 000431796000010Scopus ID: 2-s2.0-85044256972OAI: oai:DiVA.org:mdh-40323DiVA, id: diva2:1235754
Available from: 2018-07-27 Created: 2018-07-27 Last updated: 2018-09-30Bibliographically approved

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Richter, Johan

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