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: a 3 → 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 algebra m n(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 m n(f),n ≥ 3,where f is any field with ch f ≠ 2,is also zero jordan triple product determined. copyright © 2012 hongmei yao and baodong zheng.