On three-dimensional associative algebras

Abstract

This paper addresses the classification problem of associative algebras over arbitrary base fields. We present a list of three-dimensional associative algebras with canonical representatives of the isomorphism classes for fields of characteristic different from two and three. We also compare our lists with the most recent classifications over the complex numbers and with the nilpotent case over arbitrary base fields in dimension three, adding some comments.

This is a revised version of arXiv:2508.04104v3 [math.RA] 27 Aug 2025