In this paper, we generalize the Posner’s theorem on derivations in rings as follows: Let R be an arbitrary ring, P be a prime ideal of R, and d be a derivation of R. If [[x, d(x)], y] ∈ P for all x, y ∈ R, then d(R) ⊆ P or R/P is commutative. In particular, if R is semiprime and d is a centralizing derivation of R, we prove that either R is commutative or there exists a minimal prime ideal P of R such that d(R) ⊆ P. As a consequence, we show that for any semiprime ring with a centralizing derivation there exists at least a minimal prime ideal P such that d(P) ⊆ P.

中文翻译：

---

环上导数的Posner定理的推广

在本文中，我们对环上的导数进行了Posner定理的推广，如下所示：设R为任意环，P为R的素理想，而d为R的导数。如果[[ X，d（X）]，ÿ ]∈ P对所有X，ÿ ∈ [R ，则d（[R）⊆ P或- [R / P是可交换的。特别地，如果R是半素数，而d是的集中式推导[R ，证明了无论是- [R是可交换的或存在最小的素理想P的[R ，使得d（[R）⊆ P。因此，我们表明，对于一个集中推导任何半素环至少存在一个极小素理想P，使得d（P）⊆ P。