Jordan product determined points in matrix algebras

Yang Wenlei; Zhu Jun

paper

Jordan product determined points in matrix algebras

arXiv:1111.4108

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 $T : M_{n} (R)^{2} \to X$ 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.

12 pages

References in corpus (1)

Jordan higher all-derivable points in triangular algebras