# Journal of the Ramanujan Mathematical Society

Volume 39, Issue 1, March 2024 pp. 65–78.

Some simple derivations of polynomial rings

**Authors**:
Ashish Kumar Kesarwany and Vinay Wagh

**Author institution:**
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781 039.

**Summary: **
Let k be an algebraically closed field of characteristic zero. In this
article, we consider the class of derivations of k[x,y] of the form y²
∂ x +(xy+h) ∂{y}, where h ∈ k[x]. For such a derivation d,
we prove that k[x,y] is d-simple, by showing that there is no prime ideal
invariant under d.
We also prove that the derivation y∂{x} + (y² + xy + 1) ∂{y} of
k[x,y] is simple, by showing that it has no Darboux element.
Using a result by Shamsuddin [Sha77], we construct a class of simple
derivations of k[x{1}, ⋯, x{n}].

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