## Nihonkai Mathematical Journal

- Nihonkai Math. J.
- Volume 29, Number 2 (2018), 69-76.

### Notes on kernels of rational higher derivations in integrally closed domains

#### Abstract

Let $k$ be a field of characteristic $p \geq 0$ and $A = k[x_0, x_1, x_2, \ldots]$ the polynomial ring in countably many variables over $k$. We construct a rational higher $k$-derivation on $A$ whose kernel is not the kernel of any higher $k$-derivation on $A$. This example extends [5, Example 4].

#### Note

This work was supported by JSPS KAKENHI Grant Number JP17K05198.

#### Article information

**Source**

Nihonkai Math. J., Volume 29, Number 2 (2018), 69-76.

**Dates**

Received: 24 July 2018

Revised: 4 October 2018

First available in Project Euclid: 5 August 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.nihmj/1565048775

**Mathematical Reviews number (MathSciNet)**

MR3989233

**Zentralblatt MATH identifier**

07097314

**Subjects**

Primary: 13N10: Rings of differential operators and their modules [See also 16S32, 32C38]

Secondary: 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]

**Keywords**

higher derivation rational higher derivation regular field extension

#### Citation

Kojima, Hideo. Notes on kernels of rational higher derivations in integrally closed domains. Nihonkai Math. J. 29 (2018), no. 2, 69--76. https://projecteuclid.org/euclid.nihmj/1565048775