Presentations for subrings and subalgebras of finite co-rank
| Autor: | Nik Ruskuc, Peter Mayr |
|---|---|
| Přispěvatelé: | University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra, University of St Andrews. Pure Mathematics, University of St Andrews. School of Mathematics and Statistics |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Noetherian
Pure mathematics General Mathematics T-NDAS Reidemeister-Schreier K-algebra 01 natural sciences 0103 physical sciences Lie algebra FOS: Mathematics QA Mathematics Ideal (ring theory) 0101 mathematics QA Mathematics Ring Ring (mathematics) Noetherian ring Free algebra 010102 general mathematics Subalgebra Mathematics - Rings and Algebras Subring Finitely generated Finitely presented 16S15 Rings and Algebras (math.RA) 010307 mathematical physics BDC Free Lie algebra |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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