Zero action determined modules for associative algebras

Abstract

Let A be a unital associative algebra over a field F and V be a unital left A-module. The module V is called zero action determined if every bilinear map f : A × V → F with the property that f (a, m) = 0 whenever am = 0 is of the form f (x, v) = Φ(xv) for some linear map Φ : V → F . In this paper, we classify the finite dimensional irreducible and principal projective zero action determined modules of A. As an application, two classes of zero product determined algebras are shown: some semiperfect algebras (infinite dimensional in general); quasi-hereditary cellular algebras.