Algebras of Minimal Multiplicative Complexity

We prove that an associative algebra A has minimal rank if and only if the Alder-Strassen bound is also tight for the multiplicative complexity of A, that is, the multiplicative complexity of A is 2 dim A - t
_A
where t
_A
denotes the number of maximal twosided ideals of A. This generalizes a result by E. Feig who proved this for division algebras. Furthermore, we show that if A is local or superbasic, then every optimal quadratic computation for A is almost bilinear.