Maps determined by action on identity-product elements

Abstract

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