Original scientific paper

Jordan product determined points in matrix algebras

Jun Zhu ; Institute of Mathematics, Hangzhou Dianzi University, Hangzhou, P. R. China
Wenlei Yang ; Institute of Mathematics, Hangzhou Dianzi University, Hangzhou, P. R. China

Abstract

Let $M_n(R)$ be the algebra of all $n\times n$ matrices over a unital commutative ring $R$ with 6 invertible. We say that $A\in M_n(R)$ is a Jordan product determined point if for every $R$-module $X$ and every symmetric $R$-bilinear map $\{\cdot, \cdot\}$ : $M_n(R)\times M_n(R)\to X$ the following two conditions are equivalent: (i) there exists a fixed element $w\in X$ such that $\{x,y\}=w$ whenever $x\circ y=A$, $x,y\in M_n(R)$; (ii) there exists an $R$-linear map $T:M_n(R)\to X$ such that $\{x,y\}=T(x\circ y)$ for all $x,y\in M_n(R)$. In this paper, we mainly prove that all matrix units are Jordan product determined points in $M_n(R)$ when $n\geq 3$. In addition, we get some corollaries by applying the main results.

Keywords

Jordan product determined point; matrix algebra; Jordan all-multiplicative point; Jordan all-derivable point

Hrčak ID:

110821

URI

https://hrcak.srce.hr/110821

Publication date:

19.11.2013.

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