  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.
