Rings of low rank with a standard involution

TL;DR

This paper classifies low-rank R-algebras, focusing on degree 2 algebras with involutions, and identifies exceptional rings that characterize all degree 2, rank 3 algebras across various ranks.

Contribution

It provides a classification of low-rank algebras, linking degree 2 algebras with standard involutions to exceptional rings that characterize degree 2, rank 3 cases.

Findings

Algebras of degree 2 relate to standard involutions.

Exceptional rings of degree 2 occur in all ranks n ≥ 1.

These exceptional rings characterize all degree 2, rank 3 algebras.

Abstract

We consider the problem of classifying (possibly noncommutative) R-algebras of low rank over an arbitrary base ring R. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard involution. We then investigate a class of exceptional rings of degree 2 which occur in every rank n >= 1 and show that they essentially characterize all algebras of degree 2 and rank 3.