Simple algebras of Gelfand-Kirillov dimension two

TL;DR

This paper proves that finitely generated simple Goldie algebras with quadratic growth are noetherian with Krull dimension one, and explores properties and questions related to these algebras when assumptions are relaxed.

Contribution

It establishes that simple Goldie algebras of quadratic growth are noetherian with Krull dimension one, and analyzes their ideal generation properties.

Findings

Such algebras are noetherian and have Krull dimension 1.

Every ideal in these algebras is generated by at most two elements.

The paper discusses extensions and questions when assumptions are relaxed.

Abstract

Let $k$ be a field. We show that a finitely generated simple Goldie $k$-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and left ideal is generated by at most two elements. We conclude by posing questions and giving examples in which we consider what happens when the hypotheses are relaxed.