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Presentations for subrings and subalgebras of finite co-rank

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RingsFI29.pdf (301.4Kb)
Date
03/2020
Author
Mayr, Peter
Ruskuc, Nikola
Keywords
Ring
K-algebra
Finitely presented
Finitely generated
Subalgebra
Free algebra
Reidemeister-Schreier
QA Mathematics
T-NDAS
BDC
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Abstract
Let K be a commutative Noetherian ring with identity, let A be a K-algebra and let B be a subalgebra of A such that A/B is finitely generated as a K-module. The main result of the paper is that A is finitely presented (resp. finitely generated) if and only if B is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on K and show that for finite generation it can be replaced by a weaker condition that the module A/B be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension 1 of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.
Citation
Mayr , P & Ruskuc , N 2020 , ' Presentations for subrings and subalgebras of finite co-rank ' , Quarterly Journal of Mathematics , vol. 71 , no. 1 , pp. 53-71 . https://doi.org/10.1093/qmathj/haz033
Publication
Quarterly Journal of Mathematics
Status
Peer reviewed
DOI
https://doi.org/10.1093/qmathj/haz033
ISSN
0033-5606
Type
Journal article
Rights
Copyright © 2019 The Author(s). Published by Oxford University Press. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1093/qmathj/haz033
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  • University of St Andrews Research
URL
https://arxiv.org/abs/1709.04435
URI
http://hdl.handle.net/10023/21068

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