Affine representability and decision procedures for commutativity theorems for rings and algebras

TL;DR

This paper develops an algorithm based on affine representability to decide whether a set of polynomial identities enforces ring commutativity, revisiting classical theorems and characterizing identities that imply commutativity.

Contribution

It introduces a finite-step algorithm for verifying commutativity from polynomial identities, leveraging recent affine representability results.

Findings

Algorithm terminates after finite steps for given identities

Revisits and generalizes classical commutativity theorems

Characterizes multilinear identities that imply commutativity

Abstract

We consider applications of a finitary version of the Affine Representability theorem, which follows from recent work of Belov-Kanel, Rowen, and Vishne. Using this result we are able to show that when given a finite set of polynomial identities, there is an algorithm that terminates after a finite number of steps which decides whether these identities force a ring to be commutative. We then revisit old commutativity theorems of Jacobson and Herstein in light of this algorithm and obtain general results in this vein. In addition, we completely characterize the homogeneous multilinear identities that imply the commutativity of a ring.