Representability of Noetherian PI-algebras

TL;DR

This paper investigates the conditions under which Noetherian PI-algebras can be represented, extending known results and providing new machinery, while also constructing examples of non-representable cases to delineate the boundaries.

Contribution

It introduces a new representability framework for Noetherian PI-algebras containing a field and constructs non-representable examples to show the limits of these results.

Findings

Extended representability machinery for Noetherian PI-algebras containing a field.

Constructed non-representable PI-algebras demonstrating the sharpness of the results.

Connected the results to previous theorems, clarifying the scope of representability.

Abstract

This note concerns the still open question of representability of Noetherian PI-algebras. Extending a result of Rowen and Small (with an observation of Bergman) that every finitely generated module over a commutative Noetherian ring containing a field is representable, we provide a representability machinery for a Noetherian PI-algebra $R$ containing a field, which includes the case that $R$ is finite (as a module) over a commutative subalgebra isomorphic to $R/N$. We construct a family of non-representable PI-algebras demonstrating the sharpness of these results, as well as of some well known previous representability results.