Zero Triple Product Determined Matrix Algebras

Abstract

Let $A$ be an algebra over a commutative unital ring $\mathcal{C}$. We say that $A$ is zero triple product determined if for every $\mathcal{C}$-module $X$ and every trilinear map $\{\cdot ,\cdot ,\cdot \}$, the following holds: if $\{x,y,z\}=0$ whenever $xyz=0$, then there exists a $\mathcal{C}$-linear operator $T:A^3\to X$ such that $\left\{x,y,z\right\}=T\left(xyz\right)$ for all $x,y,z\in A$. 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 $M_n\left(B\right)$, $n\ge 3$, where B is any commutative unital algebra even different from the above mentioned commutative unital algebra $\mathcal{C}$, is always zero triple product determined, and $M_n\left(F\right)$, $n\ge 3$, where F is any field with ch$F\ne 2$, is also zero Jordan triple product determined.