Two-sided zero product determined algebras

An algebra $A$ is said to be two-sided zero product determined if every bilinear functional $\phi :A\times A\to F$ satisfying $\phi (x,y)=0$ whenever $xy=yx=0$ is of the form $\phi (x,y)={\tau }_1(xy)+{\tau }_2(yx)$ 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.