The prime spectrum of algebras of quadratic growth

TL;DR

This paper investigates the structure of prime algebras with quadratic growth, revealing bounds on prime ideals and properties related to matrix images, primitivity, and Jacobson radical, with implications for algebra classification.

Contribution

It establishes new bounds on prime ideals in prime monomial algebras of quadratic growth and explores their structural properties, including primitivity and radical behavior.

Findings

Prime monomial algebras of quadratic growth have finitely many prime ideals with GK dimension one.

The intersection of certain prime ideals in prime graded algebras of quadratic growth is non-empty.

Existence of a prime monomial algebra of GK dimension two with unbounded matrix images.

Abstract

We study prime algebras of quadratic growth. Our first result is that if $A$ is a prime monomial algebra of quadratic growth then $A$ has finitely many prime ideals $P$ such that $A/P$ has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the nonzero prime ideals $P$ such that $A/P$ has GK dimension 2 is non-empty, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has nonzero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra $A$ of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that $A/P$ has GK dimension 1.