Presentations for subrings and subalgebras of finite co-rank

TL;DR

This paper investigates conditions under which subrings and subalgebras of finite co-rank inherit finite presentation or generation properties, establishing equivalences and exploring limitations in non-associative contexts.

Contribution

It proves that finite co-rank subalgebras of finitely presented algebras are finitely presented, extending known results and analyzing the necessity of the Noetherian condition.

Findings

Subring of finite index in a finitely presented ring is finitely presented.

Subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented.

Results do not extend straightforwardly to non-associative algebras.

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.