PI Artin--Schelter regular algebras of dimension 3 are unique factorization rings

TL;DR

This paper proves that all noetherian PI Artin--Schelter regular algebras of dimension 3 are unique factorization rings, extending the classical UFD property to a noncommutative algebraic setting.

Contribution

It establishes the UFR property for a broad class of noncommutative algebras, linking noncommutative regularity with unique factorization.

Findings

All noetherian PI Artin--Schelter regular algebras of dimension 3 are UFRs

The result parallels the classical UFD property for regular local rings of dimension 3

Supports the broader program relating noncommutative regularity to factorization properties

Abstract

We prove that all noetherian PI Artin--Schelter regular algebras of dimension $3$ are unique factorization rings. In a certain sense, this result is a noncommutative analogue to the fact that regular local rings of dimension 3 are UFDs. The fact constitutes a crucial component in the proof of the assertion that all regular local rings are UFDs, known as the Auslander--Buchsbaum theorem.