Idempotent elements determined matrix algebras

Let Mn(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 Mn(R)×Mn(R) to V satisfies the condition that {u,u}={e,u} whenever u2=u, then there exists a linear map f from Mn(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on Mn(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on Mn(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.