Nonlinear Maps Satisfying Derivability of a Class of Matrix Ring over Commutative Rings
Received:September 17, 2014  Revised:April 25, 2015
Key Word: maps satisfying derivability   derivations   strictly upper triangular matrices   commutative rings
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11171343; 11426121) and the Science Foundation of Jiangxi University of Science and Technology (Grant Nos.NSFJ2014--K12; NSFJ2015--G24).
 Author Name Affiliation Shikun OU School of Science, Jiangxi University of Science and Technology, Jiangxi 341000, P. R. China Jin ZHONG School of Science, Jiangxi University of Science and Technology, Jiangxi 341000, P. R. China
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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.