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Israel Journal of Mathematics

, Volume 224, Issue 1, pp 263–292 | Cite as

Non-associative Ore extensions

  • Patrik Nystedt
  • Johan Öinert
  • Johan Richter
Article

Abstract

We introduce non-associative Ore extensions, S = R[X; σ, δ], for any nonassociative unital ring R and any additive maps σ, δ: RR satisfying σ(1) = 1 and δ(1) = 0. In the special case when δ is either left or right R δ -linear, where R δ = 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; id R , 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 ) = R δ [p] for a monic pR δ [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 R δ 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 R δ 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.

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Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of Engineering ScienceUniversity WestTrollhättanSweden
  2. 2.Department of Mathematics and Natural SciencesBlekinge Institute of TechnologyKarlskronaSweden
  3. 3.Academy of Education, Culture and CommunicationMälardalen UniversityVästeråsSweden

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