Zero Triple Product Determined Matrix Algebras

Let A be an algebra over a commutative unital ring C. We say that A is zero triple product determined if for every C-module X and every trilinear map {·, ·, ·}, the following holds: if {x, y, z} 0 whenever xyz 0, then there exists a C-linear operator T : A3 −→ X such that {x, y, z} T xyz for all x, y, z ∈ A. If the ordinary triple product in the aforementioned definition is replaced by Jordan triple product, then A is called zero Jordan triple product determined. This paper mainly shows that matrix algebraMn B , n ≥ 3, where B is any commutative unital algebra even different from the above mentioned commutative unital algebra C, is always zero triple product determined, and Mn F , n ≥ 3, where F is any field with chF / 2, is also zero Jordan triple product determined.