Two-sided zero product determined algebras

TL;DR

This paper characterizes finite-dimensional simple algebras that are two-sided zero product determined, showing they are exactly the separable algebras unless they are division algebras, and explores related properties and definitions.

Contribution

It provides a complete characterization of two-sided zero product determined simple algebras, linking this property to separability in finite dimensions.

Findings

Finite-dimensional simple non-division algebras are two-sided zero product determined if and only if they are separable.

The paper establishes basic properties and equivalent definitions of two-sided zero product determined algebras.

Connections between zero product properties and derivations are examined.

Abstract

An algebra $A$ is said to be two-sided zero product determined if every bilinear functional $\varphi:A\times A\to F$ satisfying $ \varphi(x,y)=0$ whenever $xy=yx=0$ is of the form $\varphi(x,y)=\tau_1(xy) + \tau_2(yx)$ for some linear functionals $\tau_1,\tau_2$ on $A$. We present some basic properties and equivalent definitions, examine connections with some properties of derivations, and as the main result prove that a finite-dimensional simple algebra that is not a division algebra is two-sided zero product determined if and only if it is separable.