## Journal of Applied Mathematics

### 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 $T:{A}^{3}{\rightarrow}X$ such that $\{x,y,z\}=T(xyz)$ 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 ${M}_{n}(B)$, $n\ge 3$, 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 ${M}_{n}(F)$, $n\ge 3$, where F is any field with ch$F\ne 2$, is also zero Jordan triple product determined.

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 925092, 18 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.jam/1355495056

Digital Object Identifier
doi:10.1155/2012/925092

Mathematical Reviews number (MathSciNet)
MR2889108

Zentralblatt MATH identifier
1239.16031

#### Citation

Yao, Hongmei; Zheng, Baodong. Zero Triple Product Determined Matrix Algebras. J. Appl. Math. 2012 (2012), Article ID 925092, 18 pages. doi:10.1155/2012/925092. https://projecteuclid.org/euclid.jam/1355495056