Nonlinear mappings on upper triangular matrices derivable at zero point

Let T(n,F)T(n,F) be the set of all n×nn×n upper triangular matrices over a field F. Recently, by using a technique from graph theory, L. Wang in [8] proved that a bijective mapping σ   on T(n,F)T(n,F), with F   a finite field, preserves zero product, i.e., σ(x)σ(y)=0σ(x)σ(y)=0 whenever xy=0xy=0, if and only if σ   can be decomposed into the product of an inner automorphism, an automorphism of T(n,F)T(n,F) induced by an automorphism of F   and a regular automorphism of the zero-divisor graph of T(n,F)T(n,F). Algebraic group theory shows that derivations of an algebra often go parallel with automorphisms of the algebra. This viewpoint encourages us to determine nonlinear mappings χ   on T(n,F)T(n,F) which are derivable at zero point, i.e., χ(x)y+xχ(y)=0χ(x)y+xχ(y)=0 whenever xy=0xy=0. It is proved that a mapping χ   on T(n,F)T(n,F) is derivable at zero point if and only if χ   is the sum of an inner derivation, an additive derivation of T(n,F)T(n,F) induced by an additive derivation of F  , and a regular mapping on T(n,F)T(n,F).