Idempotent elements determined matrix algebras

• Published in: Linear algebra and its applications Vol. 435; no. 11; pp. 2889 - 2895
• Main Authors: Wang, Dengyin, Li, Xiaowei, Ge, Hui
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.12.2011; Elsevier
• Subjects: Algebra; Bilinear maps; Exact sciences and technology; Idempotent elements determined algebras; Linear and multilinear algebra, matrix theory; Mathematics; Sciences and techniques of general use; Zero product determined algebras; Zero product determined algebras; Bilinear maps; 15A99; 15A27; 15A04; 15A03; Idempotent elements determined algebras; Automorphism; Matrix algebra; Tensor product; Commutative ring; Idempotent matrix; Jordan algebra; Commutativity; Linear transformation
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2011.05.002

Let M n ( R ) be the algebra of all n × n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map { · , · } from M n ( R ) × M n ( R ) to V satisfies the condition that { u , u } = { e , u } whenever u 2 = u , then there exists a linear map f from M n ( R ) to V such that { x , y } = f ( x ∘ y ) , ∀ x , y ∈ M n ( R ) . Applying the main result we prove that an invertible linear transformation θ on M n ( R ) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on M n ( R ) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.