Zero Triple Product Determined Matrix Algebras

Abstract EN

Let A be an algebra over a commutative unital ring ?.

We say that A is zero triple product determined if for every ?-module X and every trilinear map {·,·,·}, the following holds: if {x,y,z}=0 whenever xyz=0, then there exists a ?-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 algebra Mn(B), n≥3, where B is any commutative unital algebra even different from the above mentioned commutative unital algebra ?, 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.