Jordan product determined points in matrix algebras (Q2869098)

Let \(R\) be a commutative unital ring with the additional assumption that 6 is invertible. It is said 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 \(T:M_n(R)\to X\) such that \((x,y)=T(x\circ y)\) for all \(x,y\in M_n(R)\). Here \(x\circ y=xy+yx\) is a Jordan product. The authors prove that all matrix units are Jordan product determined points in \(M_n(R)\) for \(n\geq 3\). Several interesting corollaries are obtained.