On the Multiplicative Complexity of Modules over Associative Algebras

Abstract

The complexity $L(A,M)$ of a finite-dimensional module M over a finite-dimensional associative algebra A is the number of nonscalar multiplications/divisions of an optimal algorithm to compute the product of an element of the algebra with an element of the module.It is known that \[ L(A,A) \geqq 2 \cdot \dim A - s, \] where s is the number of maximal two-sided ideals of A. We give a generalization of this lower bound to arbitrary A-modules M.