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.