Algebras of Minimal Rank over Perfect Fields

Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the so-called Alder– Strassen boundR(A) 2 dimA t, where t is the number of maximal twosided ideals of A. The class of algebras for which the Alder–Strassen bound is sharp, the so-called algebras of minimal rank, has received a wide attention in algebraic complexity theory. We characterize all algebras of minimal rank over perfect fields. This solves an open problem in algebraic complexity theory over perfect fields, see [19, Sect. 12, Problem 4] or [9, Problem 17.5]. As a byproduct, we determine all algebras A of minimal rank with A= radA = kt over arbitrary fields.