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The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring

  • Darwin P. Mangubat
  • Jocelyn P. Vilela
  • Johan Richter
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Abstract

In this paper, we give conditions on a unital non-associative ring R and additive map σ: R⟶ R so that the non-associative skew power series ring, denoted by R[[X;σ]] and the non-associative skew Laurent series ring denoted by R((X;σ)) are left (resp. right) Noetherian, respectively. Consequently, we establish versions of Hilbert Basis Theorem of R[[X;σ]] and R((X;σ)), respectively, for left (resp. right) Noetherianity . We also give a counterexample showing that the Hilbert Basis Theorem does not hold in general in the non-associative case. In establishing the Hilbert Basis Theorems for R[[X;σ]] and R((X;σ)), we introduce and utilize the right (resp. left) quasi-associative property of R, which is equivalent to rR (resp. Rc) being one-sided ideals for all r∈ R (resp. c∈ R).

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Cite

BibTeXRIS
BibTeX
@misc{mangubat2026hilbert,
  title = {{The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring}},
  author = {Darwin P. Mangubat and Jocelyn P. Vilela and Johan Richter},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31486},
  eprint = {2609.31486},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31486},
}
RIS
TY  - UNPB
TI  - The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
AU  - Darwin P. Mangubat
AU  - Jocelyn P. Vilela
AU  - Johan Richter
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31486
UR  - https://arxiv.org/abs/2609.31486
AB  - In this paper, we give conditions on a unital non-associative ring R and additive map σ: R⟶ R so that the non-associative skew power series ring, denoted by R[[X;σ]] and the non-associative skew Laurent series ring denoted by R((X;σ)) are left (resp. right) Noetherian, respectively. Consequently, we establish versions of Hilbert Basis Theorem of R[[X;σ]] and R((X;σ)), respectively, for left (resp. right) Noetherianity . We also give a counterexample showing that the Hilbert Basis Theorem does not hold in general in the non-associative case. In establishing the Hilbert Basis Theorems for R[[X;σ]] and R((X;σ)), we introduce and utilize the right (resp. left) quasi-associative property of R, which is equivalent to rR (resp. Rc) being one-sided ideals for all r∈ R (resp. c∈ R).
ER  -