交换环上一类矩阵环上具有可导性的非线性映射

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

中文关键词:  具有可导性的映射  导子  严格上三角矩阵  交换环

英文关键词:maps satisfying derivability  derivations  strictly upper triangular matrices  commutative rings

中文摘要:

      设$R$是任意的有单位元的交换环, ${N}_n(R)$是$R$上所有$n\times n$严格上三角矩阵组成的集合. 本文详细刻画了${N}_n(R)$中所有满足$\phi(xy)=\phi(x)y+x\phi(y)$的变换$\phi$ ($\phi$ 不必具有线性或可加性的条件). 作为结论的应用, 本文对${N}_n(R)$的加法导子和导子也进行了描述.

英文摘要:

      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.