## Journal of Applied Mathematics

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 14 December 2012

**Permanent link to this document**

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