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On the differential simplicity of polynomial rings. (English) Zbl 1047.16018
The aim of the paper is to present some new examples of rings with a derivation $$d$$ having no nontrivial $$d$$-invariant ideals. Let $$\beta\in \mathbb Q[X,Y]$$ be an irreducible homogeneous polynomial of degree $$n\geqslant 3$$. Then $$\mathbb C[X,Y]$$ is $$d$$-simple with respect to the derivation $D=\left[\beta (X+Y) +b\right] \dfrac{\partial}{\partial X} +\beta \dfrac{\partial}{\partial Y}$ for any nonzero $$b\in \mathbb Q$$. The proof is based on some geometrical considerations.

##### MSC:
 16W25 Derivations, actions of Lie algebras 16S36 Ordinary and skew polynomial rings and semigroup rings 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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