关于Chuang和Lee的一个定理

A Theorem of Chuang and Lee

Chuang和Lee通过在半素环中构造一个可数子环的方法证明如下结果:设R是一个半素环,d为R上的一个导子,假设对于任意x∈R,存在一个依赖于x的多项式gx(t)∈Z(t),使得d(x-x2gx(x))=0.那么d(R)[R,R]=0.本短文指出:此定理完全可以通过常规的方法来证明.从而说明上面的定理作为Chuang和Lee方法的应用例子是不适合的.

By the method of constructing a countable subring in semiprime rings Chuang and Lee proved the following result ： Let R be a semiprime ring with d a derivation of R. Suppose that for each x R, there is a polynomial gx （t） Z [ t ], depending on x such that d （ x - X2gx （x） ） = 0. Then d （R） [ R, R ] = 0. In this short note we show that this theorem can be proved completely by a routine method. It implies that the above theorem as an application of Chuang and Lee method is not suitable.