| Title:
|
A note on the kernels of higher derivations (English) |
| Author:
|
Li, Jiantao |
| Author:
|
Du, Xiankun |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2013 |
| Pages:
|
583-588 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $k\subseteq k'$ be a field extension. We give relations between the kernels of higher derivations on $k[X]$ and $k'[X]$, where $k[X]:=k[x_1,\dots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D=\{D_n\}_{n=0}^\infty $ a higher $k$-derivation on $k[X]$ and $D'=\{D_n'\}_{n=0}^\infty $ a higher $k'$-derivation on $k'[X]$ such that $D'_m(x_i)=D_m(x_i)$ for all $m\geq 0$ and $i=1,2,\dots ,n$. Then (1) $k[X]^D=k$ if and only if $k'[X]^{D'}=k'$; (2) $k[X]^D$ is a finitely generated $k$-algebra if and only if $k'[X]^{D'}$ is a finitely generated $k'$-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. (English) |
| Keyword:
|
higher derivation |
| Keyword:
|
field extension |
| Keyword:
|
closed polynomial |
| MSC:
|
13A50 |
| idZBL:
|
Zbl 06282099 |
| idMR:
|
MR3125643 |
| DOI:
|
10.1007/s10587-013-0041-1 |
| . |
| Date available:
|
2013-10-07T11:56:14Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143478 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
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| . |
