On a class of primary algebras of minimal rank

Abstract Denote the rank ( = bilinear multiplicative complexity) of an associative k -algebra A by R ( A ). It is well known that R ( A )⩾2dim A − t , where t is the number of maximal two-sided ideals of A . A is said to be of minimal rank iff R ( A )=2dim A − t . This paper is concerned with the structure of primary algebras ( t =1) of minimal rank. We show that a primary k -algebra is of minimal rank if there exists a pair (1, x 2 ,…, x n ), (1, y 2 ,…, y n ) of k -bases of A such that for all i , j , x i y j ∈ ky i + ky j . At least for local algebras, this sufficient condition is also necessary and leads to a complete description of the structure of local algebras of minimal rank. As an application we investigate the ranks of the 5-dimensional local algebras. Furthermore it turns out that k 2×2 is the only primary, nonlocal algebra satisfying the abovementioned sufficient condition.