Two-sided zero product determined triangular algebras.

Let 풰 be an algebra over the field 픽. We say that 풰 is two-sided zero product determined if every bilinear functional ϕ : 풰×풰→ 픽 the following holds: if ϕ(x,y) = 0 whenever xy = yx = 0, then there exist linear functionals F1 and F2 on 풰 such that ϕ(x,y) = F1(xy)+F2(yx) for all x,y ∈풰. We show that the unital triangular algebra 풯 = 풜ℳ0 ℬ is a two-sided zero product determined algebra if and only if 풜 and ℬ 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. [ABSTRACT FROM AUTHOR]