Nonlinear Maps Satisfying Derivability of a Class of Matrix Ring over Commutative Rings

Abstract:

      Let $R$ be an arbitrary commutative ring with identity, and let ${N}_n(R)$ be the set consisting of all $n\times n$ strictly upper triangular matrices over $R$. In this paper, we give an explicit description of the maps (without linearity or additivity assumption) $\phi:{N}_n(R)\rightarrow {N}_n(R)$ satisfying $\phi(xy)=\phi(x)y+x\phi(y)$. As a consequence, additive derivations and derivations of ${N}_n(R)$ are also described.