交换环上一类代数的拟导子

Quasi-derivations of a Algebra over Commutative Ring

设R为任意含幺交换环,Mn(R)为R上所有矩阵组成的结合R-代数。对于Mn(R)上线性变换φ,若存在线性变换φ′使得对任意x,y∈Mn(R)均有φ′xy=φxy+xφy,则称φ为Mn(R)上的拟导子。本文定出了当n≥3时Mn(R)上任一拟导子的具体形式,对导子的概念进行了推广。

Let R be an arbitrary commutative ring with identity. Denote by M （R） the associative R -algebra over R consisting of all n by n matrices. An invertible linear transformation φ on Mn （R） is called a quasi-derivation of it if there exists an invertible linear transformation φ＇ on Mn（R） such that φ＇（xy） = φ（x）y ＋ xφ（y） for Vx,y ∈ Mn（R）. The aim of this paper is to give an explicit description on the quasi-derivations of Mo （R） when n≥ 3. Generalizes the notion of derivation to a more general case.