Finite dimensional zero product determined algebras are generated by idempotents (Q5965386)

An algebra \(A\) is said to be zero product determined if every bilinear map \(f\) from \(A\times A\) into an arbitrary vector space \(X\) with the property that \(f(x,y)=0\) whenever \(xy=0\) is of the form \(f(x,y)=\Phi(xy)\) for some linear map \(\Phi:A\to X\).NEWLINENEWLINELet \(A\) be an algebra. If the Jacobson radical of \(A\) has codimension \(1\) in \(A\) and is nonzero, then \(A\) is not zero product determined. Also, if \(A\) has an ideal \(I\) of codimension \(1\) such that \(I^{2}\not=I\), then \(A\) is not zero product determined. Another example of an algebra which is not zero product determined is the direct product \(F\times F\times\cdots\) of countably infinitely many copies of an infinite field \(F\).NEWLINENEWLINEThe main result of this paper states that a finite dimensional (unital) algebra is zero product determined if and only if it is generated by idempotents.