Maps determined by action on identity-product elements

Let Mn(R) be the algebra of all n×n matrices over a unital commutative ring R with 6 invertible. For a given element z∈Mn(R), a map δ on Mn(R) is called preserving z-product if δ(x)δ(y)=δ(z) whenever xy=z. A map σ on Mn(R) is called derivable at the given point z if σ(x)y+xσ(y)=σ(z) whenever xy=z. Using elementary matrix technique we show that if an invertible linear map δ on Mn(R) preserves identity-product, then it is a Jordan automorphism; and a linear map σ on Mn(R) is derivable at the identity matrix if and only if it is an inner derivation.