Zero action determined modules for associative algebras

TL;DR

This paper classifies certain modules over associative algebras that are characterized by a specific zero-action property, and identifies classes of algebras with this property, expanding understanding of module and algebra structures.

Contribution

It provides a classification of finite dimensional irreducible and principal projective zero action determined modules and identifies classes of zero product determined algebras.

Findings

Classification of finite dimensional irreducible zero action determined modules

Identification of zero product determined classes among semiperfect and quasi-hereditary cellular algebras

Extension of zero action concepts to infinite dimensional cases

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.