Jordan product determined points in matrix algebras

2011/11/17 by Yang Wenlei, Wenlei, Yang, Zhu Jun +1
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA) #math.OA #math.RA

paper · pdf · doi:10.48550/arxiv.1111.4108

12 pages

arxiv created 2011/11/17 · arxiv updated 2011/11/18

Abstract

Let M_n(R) be the algebra of all n× n matrices over a unital commutative ring R with 6 invertible. We say that A∈ M_n(R) is a Jordan product determined point if for every R-module X and every symmetric R-bilinear map \⋅, ⋅\ : M_n(R)× M_n(R)→ X the following two conditions are equivalent: (i) there exists a fixed element w∈ X such that \x,y\=w whenever x∘ y=A, x,y∈ M_n(R); (ii) there exists an R-linear map T:M_n(R)²→ X such that \x,y\=T(x∘ y) for all x,y∈ 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≥ 3. In addition, we get some corollaries by applying the main results.

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