On nonmatrix varieties of associative rings

TL;DR

This paper investigates nonmatrix varieties of associative rings over commutative rings, extending known results from infinite fields to more general base rings and varieties excluding matrix algebras.

Contribution

It generalizes existing results on nonmatrix varieties from infinite fields to arbitrary commutative rings and broader algebraic settings.

Findings

Extended known results to arbitrary commutative rings

Generalized results to varieties not containing matrix algebras

Provided new insights into the structure of nonmatrix varieties

Abstract

We study nonmatrix varieties of $\mathbf{k}$-algebras, where $\mathbf{k}$ is a unital commutative ring. Our results extend to this generality known results for the case in which $\mathbf{k}$ is an infinite field. Also, we generalize these results to varieties of $\mathbf{k}$-algebras not containing the algebra of $n\times n$ matrices.