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 $f: A\times V\rightarrow F$ with the property that $f(a,m)=0$ whenever $am=0$ is of the form $f(x,v)=\Phi(xv)$ for some linear map $\Phi: V\rightarrow F$. 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.