Prime affine algebras of GK dimension two which are almost PI algebras

TL;DR

This paper investigates prime affine algebras of Gelfand-Kirillov dimension two that are almost PI, providing conditions under which such algebras are primitive and analyzing their centers, partially addressing a longstanding question.

Contribution

It introduces conditions making certain affine prime algebras primitive and explores their properties, including their centers, in the context of almost PI algebras.

Findings

Such algebras are prime and have countable cofinal subsets of ideals.

Under additional conditions, these algebras are primitive.

The center is a finite dimensional field extension of the base field.

Abstract

An almost PI algebra is a generalisation of a just infinite algebra which does not satisfy a polynomial identity. An almost PI algebra has some nice properties: It is prime, has a countable cofinal subset of ideals and when satisfying ACC(semiprimes), it has only countably many height 1 primes. Consider an affine prime Goldie non-simple non-PI $k$-algebra $R$ of GK dimension $<3$, where $k$ is an uncountable field. $R$ is an almost PI algebra. We give some possible additional conditions which make such an algebra primitive. This gives a partial answer to Small's question: Let $R$ be an affine prime Noetherian $k$-algebra of GK dimension 2, where $k$ is any field. Does it follow that $R$ is PI or primitive? We also show that the center of $R$ is a finite dimensional field extension of $k$, and if, in addition, $k$ is algebraically closed, then $R$ is stably almost PI.