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arXiv 1111.4108math.OAmath.RA
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矩阵代数中的Jordan积决定点

Jordan product determined points in matrix algebras

Yang Wenlei, Zhu Jun

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12 pages
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摘要：

设$M_n(R)$是具有6可逆单位交换环$R$上所有$n\times n$矩阵的代数.我们说$A\in M_n(R)$是Jordan积决定点，如果对每个$R$ -模$X$和每个对称$R$ -双线性映射$\{\cdot, \cdot\}$：$M_n(R)\times M_n(R)\to X$下列两个条件是等价的：（i）存在一个固定元素$w\in X$，使得$\{x,y\}=w$每当$x\circ y=A$，$x,y\in M_n(R)$ ;（ii）存在一个$R$ -线性映射$T:M_n(R)^2\to X$，使得$\{x,y\}=T(x\circ y)$对于所有$x,y\in M_n(R)$ .本文主要证明了当$n\geq 3$ .此外，应用主要结果得到了一些推论。

英文摘要：

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 $T:M_n(R)^2\to X$ such that $\{x,y\}=T(x\circ y)$ for all $x,y\in M_n(R)$. In this paper, we mainly prove that all the matrix units are the Jordan product determined points in $M_n(R)$ when $n\geq 3$. In addition, we get some corollaries by applying the main results.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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