Jordan product determined points in matrix algebras

TL;DR

This paper characterizes matrix units in algebra of matrices over a ring as Jordan product determined points, establishing conditions under which symmetric bilinear maps depend linearly on the Jordan product.

Contribution

It proves that all matrix units in $M_n(R)$ are Jordan product determined points for $n \\geq 3$, extending understanding of bilinear maps in matrix algebras.

Findings

Matrix units are Jordan product determined points for $n \\geq 3$.

Conditions for symmetric bilinear maps to depend linearly on Jordan product.

Corollaries derived from main results.

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.