Two-Sided Zero Product Determined Triangular Algebras

Let Formula: see text be an algebra over the field Formula: see text. We say that Formula: see text is two-sided zero product determined if every bilinear functional Formula: see text the following holds: if Formula: see text whenever Formula: see text, then there exist linear functionals Formula: see text and Formula: see text on Formula: see text such that Formula: see text for all Formula: see text. We show that the unital triangular algebra Formula: see text is a two-sided zero product determined algebra if and only if Formula: see text and Formula: see text are two-sided zero product determined algebras, and then we get various results about this property for generalized triangular algebras and block upper triangular matrix algebras. We also provide an application of the main result to determine the structure of commutativity preserving maps at commutative zero products on triangular algebras. We note that some of the previous results about the two-sided zero product determined property have been generalized.