Finite dimensional zero product determined algebras are generated by idempotents

An algebra AA is said to be zero product determined if every bilinear map ff from A×AA×A into an arbitrary vector space XX with the property that f(x,y)=0f(x,y)=0 whenever xy=0xy=0 is of the form f(x,y)=Φ(xy)f(x,y)=Φ(xy) for some linear map Φ:A→XΦ:A→X. It is known, and easy to see, that an algebra generated by idempotents is zero product determined. The main new result of this partially expository paper states that for finite dimensional (unital) algebras the converse is also true. Thus, if such an algebra is zero product determined, then it is generated by idempotents.