Presentations for subrings and subalgebras of finite co-rank

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.