Algebras in which every subalgebra is noetherian

TL;DR

This paper demonstrates that certain algebraic structures derived from elliptic curves and Sklyanin algebras have the property that all their subalgebras are finitely generated and noetherian, revealing a surprising uniformity.

Contribution

It establishes that twisted homogeneous coordinate rings of elliptic curves with infinite automorphisms, and localizations of generic Sklyanin algebras, have all subalgebras finitely generated and noetherian.

Findings

All subalgebras of these rings are finitely generated.

Every subalgebra is noetherian.

This property is shown for specific classes of algebras.

Abstract

We show that the twisted homogeneous coordinate rings of elliptic curves by infinite order automorphisms have the curious property that every subalgebra is both finitely generated and noetherian. As a consequence, we show that a localisation of a generic Skylanin algebra has the same property.