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 x y z = 0 , then there exists a C -linear operator T : A 3 ⟶ X such that x , y , z = T ( x y z ) 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.