Zero Triple Product Determined Matrix Algebras

Abstract

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 algebra Mn(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.