Hopfian and Bassian algebras

TL;DR

This paper explores Hopfian and Bassian properties of rings, establishing conditions under which certain classes of algebras exhibit these properties, linking algebraic structure to chain conditions and representability.

Contribution

It characterizes classes of Hopfian and Bassian rings, connecting these properties to chain conditions and algebraic representability, and provides specific results for semiprime and affine PI-algebras.

Findings

Semiprime algebras with ACC on semiprime ideals are Hopfian.

Semiprime affine PI-algebras over a field are Bassian.

Abstract

A ring $A$ is called Hopfian if $A$ cannot be isomorphic to a proper homomorphic image $A/J$. $A$ is called Bassian if there cannot be an injection of $A$ into a proper homomorphic image $A/J$. We consider classes of Hopfian and Bassian rings, and tie representability of algebras and chain conditions on ideals to these properties. In particular, any semiprime algebra satisfying the ACC on semiprime ideals is Hopfian, and any semiprime affine PI-algebra over a field is Bassian.