# A note on the kernels of higher derivations

• Volume: 63, Issue: 3, page 583-588
• ISSN: 0011-4642

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## Abstract

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Let $k\subseteq {k}^{\text{'}}$ be a field extension. We give relations between the kernels of higher derivations on $k\left[X\right]$ and ${k}^{\text{'}}\left[X\right]$, where $k\left[X\right]:=k\left[{x}_{1},\cdots ,{x}_{n}\right]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D={\left\{{D}_{n}\right\}}_{n=0}^{\infty }$ a higher $k$-derivation on $k\left[X\right]$ and ${D}^{\text{'}}={\left\{{D}_{n}^{\text{'}}\right\}}_{n=0}^{\infty }$ a higher ${k}^{\text{'}}$-derivation on ${k}^{\text{'}}\left[X\right]$ such that ${D}_{m}^{\text{'}}\left({x}_{i}\right)={D}_{m}\left({x}_{i}\right)$ for all $m\ge 0$ and $i=1,2,\cdots ,n$. Then (1) $k{\left[X\right]}^{D}=k$ if and only if ${k}^{\text{'}}{\left[X\right]}^{{D}^{\text{'}}}={k}^{\text{'}}$; (2) $k{\left[X\right]}^{D}$ is a finitely generated $k$-algebra if and only if ${k}^{\text{'}}{\left[X\right]}^{{D}^{\text{'}}}$ is a finitely generated ${k}^{\text{'}}$-algebra. Furthermore, we also show that the kernel $k{\left[X\right]}^{D}$ of a higher derivation $D$ of $k\left[X\right]$ can be generated by a set of closed polynomials.

## How to cite

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Li, Jiantao, and Du, Xiankun. "A note on the kernels of higher derivations." Czechoslovak Mathematical Journal 63.3 (2013): 583-588. <http://eudml.org/doc/260598>.

@article{Li2013,
abstract = {Let $k\subseteq k^\{\prime \}$ be a field extension. We give relations between the kernels of higher derivations on $k[X]$ and $k^\{\prime \}[X]$, where $k[X]:=k[x_1,\dots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D=\lbrace D_n\rbrace _\{n=0\}^\infty$ a higher $k$-derivation on $k[X]$ and $D^\{\prime \}=\lbrace D_n^\{\prime \}\rbrace _\{n=0\}^\infty$ a higher $k^\{\prime \}$-derivation on $k^\{\prime \}[X]$ such that $D^\{\prime \}_m(x_i)=D_m(x_i)$ for all $m\ge 0$ and $i=1,2,\dots ,n$. Then (1) $k[X]^D=k$ if and only if $k^\{\prime \}[X]^\{D^\{\prime \}\}=k^\{\prime \}$; (2) $k[X]^D$ is a finitely generated $k$-algebra if and only if $k^\{\prime \}[X]^\{D^\{\prime \}\}$ is a finitely generated $k^\{\prime \}$-algebra. Furthermore, we also show that the kernel $k[X]^D$ of a higher derivation $D$ of $k[X]$ can be generated by a set of closed polynomials.},
author = {Li, Jiantao, Du, Xiankun},
journal = {Czechoslovak Mathematical Journal},
keywords = {higher derivation; field extension; closed polynomial; higher derivation; field extension; closed polynomial},
language = {eng},
number = {3},
pages = {583-588},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the kernels of higher derivations},
url = {http://eudml.org/doc/260598},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Li, Jiantao
AU - Du, Xiankun
TI - A note on the kernels of higher derivations
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 583
EP - 588
AB - Let $k\subseteq k^{\prime }$ be a field extension. We give relations between the kernels of higher derivations on $k[X]$ and $k^{\prime }[X]$, where $k[X]:=k[x_1,\dots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D=\lbrace D_n\rbrace _{n=0}^\infty$ a higher $k$-derivation on $k[X]$ and $D^{\prime }=\lbrace D_n^{\prime }\rbrace _{n=0}^\infty$ a higher $k^{\prime }$-derivation on $k^{\prime }[X]$ such that $D^{\prime }_m(x_i)=D_m(x_i)$ for all $m\ge 0$ and $i=1,2,\dots ,n$. Then (1) $k[X]^D=k$ if and only if $k^{\prime }[X]^{D^{\prime }}=k^{\prime }$; (2) $k[X]^D$ is a finitely generated $k$-algebra if and only if $k^{\prime }[X]^{D^{\prime }}$ is a finitely generated $k^{\prime }$-algebra. Furthermore, we also show that the kernel $k[X]^D$ of a higher derivation $D$ of $k[X]$ can be generated by a set of closed polynomials.
LA - eng
KW - higher derivation; field extension; closed polynomial; higher derivation; field extension; closed polynomial
UR - http://eudml.org/doc/260598
ER -

## References

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1. Arzhantsev, I. V., Petravchuk, A. P., 10.1007/s11253-008-0037-4, Ukr. Math. J. 59 (2007), 1783-1790. (2007) Zbl1164.13302MR2411588DOI10.1007/s11253-008-0037-4
2. Kojima, H., Wada, N., 10.1080/00927871003660200, Commun. Algebra 39 (2011), 1577-1582. (2011) Zbl1235.13023MR2821493DOI10.1080/00927871003660200
3. Mirzavaziri, M., 10.1080/00927870902828751, Commun. Algebra 38 (2010), 981-987. (2010) Zbl1191.16040MR2650383DOI10.1080/00927870902828751
4. Miyanishi, M., Lectures on Curves on Rational and Unirational Surfaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics Berlin, Springer (1978). (1978) Zbl0425.14008MR0546289
5. Nowicki, A., Polynomial Derivations and their Rings of Constants, N. Copernicus Univ. Press Toruń (1994). (1994) Zbl1236.13023MR2553232
6. Roman, S., Advanced Linear Algebra, 3rd edition, Graduate Texts in Mathematics 135 New York, Springer (2008). (2008) Zbl1132.15002MR2344656
7. Tanimoto, R., 10.1016/j.jpaa.2008.03.006, J. Pure Appl. Algebra 212 (2008), 2284-2297. (2008) Zbl1157.13004MR2426508DOI10.1016/j.jpaa.2008.03.006
8. Wada, N., 10.4064/cm122-2-3, Colloq. Math. 122 (2011), 185-189. (2011) Zbl1213.13039MR2775166DOI10.4064/cm122-2-3

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