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.
John Voight. "Rings of low rank with a standard involution." Illinois J. Math. 55 (3) 1135 - 1154, Fall 2011. https://doi.org/10.1215/ijm/1369841800