## Notes on kernels of rational higher derivations in integrally closed domains.(English)Zbl 1419.13049

Summary: 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 [P. Jȩdrzejewicz, Colloq. Math. 99, No. 1, 51–53 (2004; Zbl 1075.13504); Example 4].

### MSC:

 13N10 Commutative rings of differential operators and their modules 13F20 Polynomial rings and ideals; rings of integer-valued polynomials

Zbl 1075.13504
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### References:

 [1] A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Math., vol. 190, Birkhäuser Verlag, Basel, 2000. · Zbl 0962.14037 [2] G. Freudenburg, Algebraic theory of locally nilpotent derivations (second edition), Encyclopedia of Mathematical Sciences vol. 136, Invariant Theory and Algebraic Transformation Groups VII, Springer-Verlag, 2017. · Zbl 1391.13001 [3] N. Heerema and J. Deveney, Galois theory for fields $$K/k$$ finitely generated, Trans. Amer. Math. Soc. 189 (1974), 263-274. · Zbl 0282.12102 [4] Y. Hirano, On a theorem of Ayad and Ryckelynck, Comm. Algebra 33 (2005), 897-898. · Zbl 1065.13009 [5] P. Jędrzejewicz, A note on characterizations of rings of constants with respect to derivations, Colloq. Math. 99 (2004), 51-53. [6] H. Kojima, On the kernels of some higher derivations in polynomial rings, J. Pure Appl. Algebra 215 (2011), 2512-2514. · Zbl 1221.13041 [7] H. Kojima, Notes on the kernels of locally finite higher derivations in polynomial rings, Comm. Algebra 44 (2016), 1924-1930. · Zbl 1344.13008 [8] H. Kojima and T. Nagamine, Closed polynomials in polynomial rings over integral domains, J. Pure Appl. Algebra 219 (2015), 5493-5499. · Zbl 1320.13010 [9] H. Kojima and N. Wada, Kernels of higher derivations in $$R[x,y]$$, Comm. Algebra 39 (2011), 1577-1582. · Zbl 1235.13023 [10] A. Nowicki, Rings and fields of constants for derivations in characteristic zero, J. Pure Appl. Algebra 96 (1994), 47-55. · Zbl 0811.12003 [11] A. Nowicki, Polynomial derivations and their rings of constants, Uniwersytet Mikolaja Kopernika, Toruń, 1994. · Zbl 1236.13023 [12] S. Suzuki, Some types of derivations and their applications to field theory, J. Math. Kyoto Univ. 21 (1981), 375-382. · Zbl 0496.12018
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