多项式环上带权的单项式导子

Monomial Derivations with Weight on Polynomial Rings

设k是特征为零的域,k[x]为k上的多项式环,给出了k[x]上带权单项式导子的概念,然后通过对权是否为零进行分类讨论,证明了D是权为零的非零单项式导子当且仅当存在b∈k\{0},s∈N,使得对任意n∈N都有d(x^n)=nbx^(s+n-1);D是权为λ≠0的非零单项式导子当且仅当存在a∈k\{0},使得D(x^n)=0,n={0λ^(-1)((λa+1)~n-1)x^n,n≥1。

Monomial derivation with weight on k[x],the polynomial ring over characteristic zero filed k,is defined.Then the classification of this kind of derivations is established by discussing whether the weight is zero.More precisely,it is showed that Dis a nonzero monomial derivation with weight zero if and only if there exist b∈k／{0},s∈N,such that d（x-n）=nbx-（s＋n-1） for all n∈N;It is also proved that Dis a nonzero monomial derivation with weight λ≠0 if and only if there exists a∈k／{0} such that D（x-n）=0,n={0λ-（-1）（（λa＋1）-n-1）x-n,n≥1