A dichotomy result for prime algebras of Gelfand-Kirillov dimension two

TL;DR

This paper proves that finitely generated prime Goldie algebras over an uncountable field with quadratic growth are either primitive or satisfy a polynomial identity, resolving a previously open question.

Contribution

It establishes a dichotomy for prime algebras of Gelfand-Kirillov dimension two, showing they are either primitive or PI, which was previously unknown.

Findings

Prime Goldie algebras of quadratic growth are either primitive or satisfy a polynomial identity.

Answers a question posed by Small regarding the structure of such algebras.

Provides a clear classification for prime algebras of Gelfand-Kirillov dimension two.

Abstract

Let $k$ be an uncountable field. We show that a finitely generated prime Goldie $k$-algebra of quadratic growth is either primitive or satisfies a polynomial identity, answering a question of Small in the affirmative.