LetR be a prime ring of characteristic different from 2,U the Utumi quotient ring ofR, C the extended centroid ofR, F andG non-zero generalized derivations ofR andf(x ₁, ...,x _n ) a polynomial overC. Denote byf(R) the set {f(r₁, ..., r_n): r₁, ..., r_n ∃ R} of all the evaluations off(x ₁, ...,x _n ) inR. Suppose thatf(x ₁, ...,x _n ) is not central valued onR. IfR does not embed inM ₂(K), the algebra of 2 × 2 matrices over a fieldK, and the composition (FG) acts as a generalized derivation on the elements off(R), then (FG) is a generalized derivation of R and one of the following holds:

1. 1. there existsα ∈ C such thatF(x)=αx, for allx ∈ R;
2. 2. there existsα ∈ C such thatG(x)=αx, for allx ∈ R;
3. 3. there exista; b ∈ U such thatF(x)=ax, G(x)=bx, for allx ∈ R;
4. 4. there exista; b ∈ U such thatF(x)=xa, G(x)=xb, for allx ∈ R;
5. 5. there exista; b ∈ U, α,β ∈ C such thatF(x)=ax+xb, G(x)=αx+β(αx − xb), for allx ∈ R.

中文翻译：

---

素环上多项式的两个广义导数的乘积

令R为特性不同于2的素环，U为R，U的Utumi商环，C为R，F和G的扩展质心R和f的非零广义导数（x ₁，...，x _n）C的多项式。表示由F（R）的集合{ ˚F（R ₁，...，R _Ñ）：R ₁，...，R _Ñ ∃R}的所有评估的˚F（X ₁，...，X _Ñ）在 [R 。假设f（x ₁，...，x _n）不是R的中心值。如果R不嵌入M ₂（K）中，则在场K上2×2矩阵的代数，并且合成（FG）作为f（R）元素的广义导数，则（FG）为a R的广义推导和下列其中一项：

1. 1.对于所有x∈R， 存在α∈C使得F（x）=αx；
2. 2.对于所有x∈R， 存在α∈C使得G（x）=αx；
3. 3. 存在一个; b∈U使得F（x）= ax，G（x）= bx，对于所有x∈R ;
4. 4. 存在一个；b∈U使得F（x）= xa，G（x）= xb，对于所有x∈R ;
5. 5. 存在一个; B∈U，α，β∈Ç使得F（X）= AX + XB，G（X）=αX+β（αX - XB），对于所有的X∈R。