Zero product determined triangular algebras

Abstract

Let 𝒜 be an algebra over a commutative unital ring 𝒞. We say that 𝒜 is zero product determined if for every 𝒞-module 𝒱 and every 𝒞-bilinear map φ: 𝒜 × 𝒜 → 𝒱 the following holds: if φ(A, B) = 0 whenever AB = 0, then there exists a 𝒞-linear map L such that φ(A, B) = L(AB) for all A, B ∈ 𝒜. If we replace, in this definition, the ordinary product by the Jordan (resp. Lie) product, then we say that 𝒜 is zero Jordan (resp. Lie) product determined. We show that the triangular algebra is zero (resp. Lie) product determined if and only if 𝒜 and ℬ are zero (resp. Lie) product determined, and under some technical restrictions, a same result is true for the Jordan product. The main result is then applied to generalized triangular matrix algebras and (block) upper triangular matrix algebras. In particular, we show that: (i) the block upper triangular matrix algebra (n ≥ 1) is a zero product determined algebra; (ii) if 𝒞 contains the element , then (n ≥ 1) is a zero Jordan product determined algebra and (iii) if 𝒜 is a commutative algebra, then (n ≥ 1) is a zero Lie product determined algebra.