Near-Trinions: Complete Classification of Unital Three-Dimensional Real Associative Algebras

Abstract

Frobenius' theorem settles, definitively, the question of three-dimensional real division algebras: there are none. What it leaves open -- and what this paper addresses -- is the question of which three-dimensional real associative algebras actually exist. We call these \emph{near-trinions} and classify them completely, up to $\mathbb{R}$-algebra isomorphism. The answer is exactly six isomorphism classes; we provide canonical representatives, explicit multiplication tables, and a collection of module-theoretic invariants sufficient to distinguish every pair. The classification proceeds by stratifying on the radical dimension $d$: Wedderburn--Artin resolves the semisimple stratum; in the $d=1$ stratum, the Peirce decomposition locates the radical generator in either a diagonal or off-diagonal corner, producing the commutative and non-commutative branches respectively and ruling out the residue field $\mathbb{C}$ by a characteristic sign obstruction; and the $d=2$ stratum splits on whether Jac$(A)^2$ vanishes. The argument not only recovers the six classes but explains why the radical dimension takes values in $\{0,1,2\}$, why the $d=1$ case splits on commutativity, and why no near-trinion with residue field $\mathbb{C}$ can exist. One direction for subsequent work is proposed.