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Hilbert's Basis Theorem for Non-associative and Hom-associative Ore Extensions
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0002-6309-8709
Blekinge Tekniska Högskola.ORCID iD: 0000-0003-3931-7358
2023 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 26, p. 1051-1065Article in journal (Refereed) Published
Abstract [en]

We prove a hom-associative version of Hilbert’s basis theorem, which includes as special cases both a non-associative version and the classical Hilbert’s basis theorem for associative Ore extensions. Along the way, we develop hom-module theory. We conclude with some examples of both non-associative and hom-associative Ore extensions which are all noetherian by our theorem.

Place, publisher, year, edition, pages
Springer Science+Business Media B.V., 2023. Vol. 26, p. 1051-1065
Keywords [en]
Hilbert’s basis theorem, hom-associative algebras, hom-associative Ore extensions, hom-modules, non-commutative noetherian rings
National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-47518DOI: 10.1007/s10468-022-10123-8ISI: 000784962700001Scopus ID: 2-s2.0-85128357281OAI: oai:DiVA.org:mdh-47518DiVA, id: diva2:1423661
Available from: 2020-04-15 Created: 2020-04-15 Last updated: 2023-12-07Bibliographically approved
In thesis
1. On Hom-associative Ore Extensions
Open this publication in new window or tab >>On Hom-associative Ore Extensions
2022 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we introduce and study hom-associative Ore extensions. These are non-unital, non-associative, non-commutative polynomial rings in which the associativity condition is “twisted” by an additive group homomorphism. In particular, they are examples of hom-associative algebras, and they generalize the classical non-commutative polynomial rings introduced by Ore known as Ore extensions to the non-unital, hom-associative setting. At the same time, when the twisted associativity condition is null, they also generalize to the general non-unital, non-associative setting. We deduce necessary and sufficient conditions for hom-associative Ore extensions to exist, and construct concrete examples thereof. These include hom-associative generalizations of the quantum plane, the universal enveloping algebra of the two-dimensional non-abelian Lie algebra, and the first Weyl algebra, to name a few. The aforementioned algebras turn out to be formal hom-associative deformations of their associative counterparts, the latter two which cannot be formally deformed in the associative setting. Moreover, these are all weakly unital algebras, and we provide a way of embedding any multiplicative, non-unital hom-associative algebra into a multiplicative, weakly unital hom-associative algebra. This generalizes the classical unitalization of non-unital, associative algebras. We then study the hom-associative Weyl algebras in arbitrary characteristic, classify them up to isomorphism, and in the zero characteristic case, we prove that an analogue of the Dixmier conjecture is true. We also study hom-modules over hom-associative rings, and by doing so, we are able to prove a Hilbert's basis theorem for hom-associative Ore extensions. Our theorem includes as special cases both the classical Hilbert's basis theorem for Ore extensions and a Hilbert's basis theorem for unital, non-associative Ore extensions. Last, we construct examples of both hom-associative and non-associative Ore extensions which are all Noetherian by our theorem.

Place, publisher, year, edition, pages
Västerås: Mälardalens universitet, 2022
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 360
National Category
Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-58107 (URN)978-91-7485-553-1 (ISBN)
Public defence
2022-06-10, Kappa, Mälardalens universitet, Västerås, 13:15 (English)
Opponent
Supervisors
Available from: 2022-05-03 Created: 2022-04-25 Last updated: 2023-04-11Bibliographically approved

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Bäck, Per

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