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Journal articles on the topic 'Locally nilpotent derivations'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Karaś, Marek. "Locally Nilpotent Monomial Derivations." Bulletin of the Polish Academy of Sciences Mathematics 52, no. 2 (2004): 119–21. http://dx.doi.org/10.4064/ba52-2-2.

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2

LEVCHUK, VLADIMIR M., and OKSANA V. RADCHENKO. "DERIVATIONS OF THE LOCALLY NILPOTENT MATRIX RINGS." Journal of Algebra and Its Applications 09, no. 05 (2010): 717–24. http://dx.doi.org/10.1142/s0219498810004154.

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Derivations of the ring of all finitary niltriangular matrices over an arbitrary associative ring with identity for any chain of matrix indices are described. Every Lie or Jordan derivation is a derivation of this ring modulo third hypercenter.

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3

Finston, David R., and Sebastian Walcher. "Centralizers of locally nilpotent derivations." Journal of Pure and Applied Algebra 120, no. 1 (1997): 39–49. http://dx.doi.org/10.1016/s0022-4049(96)00064-3.

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4

El Kahoui, M’hammed. "Subresultants and locally nilpotent derivations." Linear Algebra and its Applications 380 (March 2004): 253–61. http://dx.doi.org/10.1016/j.laa.2003.11.004.

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5

Tanaka, Mikiya. "Locally nilpotent derivations on modules." Journal of Mathematics of Kyoto University 49, no. 1 (2009): 131–59. http://dx.doi.org/10.1215/kjm/1248983033.

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6

Matveev, D. A. "Commuting homogeneous locally nilpotent derivations." Sbornik: Mathematics 210, no. 11 (2019): 1609–32. http://dx.doi.org/10.1070/sm9132.

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7

BERGEN, JEFFREY, and PIOTR GRZESZCZUK. "GK DIMENSION AND LOCALLY NILPOTENT SKEW DERIVATIONS." Glasgow Mathematical Journal 57, no. 3 (2014): 555–67. http://dx.doi.org/10.1017/s0017089514000482.

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AbstractLet A be a domain over an algebraically closed field with Gelfand–Kirillov dimension in the interval [2,3). We prove that if A has two locally nilpotent skew derivations satisfying some natural conditions, then A must be one of five algebras. All five algebras are Noetherian, finitely generated, and have Gelfand–Kirillov dimension equal to 2. We also obtain some results comparing the Gelfand–Kirillov dimension of an algebra to its subring of invariants under a locally nilpotent skew derivation.

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8

Zhao, Wenhua. "Some open problems on locally finite or locally nilpotent derivations and ℰ-derivations". Communications in Contemporary Mathematics 20, № 04 (2018): 1750056. http://dx.doi.org/10.1142/s0219199717500560.

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Let [Formula: see text] be a commutative ring and [Formula: see text] an [Formula: see text]-algebra. An [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is an [Formula: see text]-linear map of the form [Formula: see text] for some [Formula: see text]-algebra endomorphism [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the identity map of [Formula: see text]. In this paper, we discuss some open problems on whether or not the image of a locally finite (LF) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation

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9

Kuroda, Shigeru. "Van den Essen's conjecture on the kernel of a derivation having a slice." Journal of Algebra and Its Applications 14, no. 09 (2015): 1540003. http://dx.doi.org/10.1142/s0219498815400034.

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The problem of finite generation of the kernel of a derivation of a polynomial ring is a special case of Hilbert's Fourteenth Problem. It is well known that the answer is affirmative if the derivation is locally nilpotent and having a slice. Van den Essen (1995) conjectured that there exists a counterexample for non-locally nilpotent derivations with a slice. In this paper, we solve this conjecture in the affirmative.

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10

El Kahoui, M’hammed, and Mustapha Ouali. "Locally nilpotent derivations of factorial domains." Journal of Algebra and Its Applications 18, no. 12 (2019): 1950222. http://dx.doi.org/10.1142/s0219498819502220.

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Let [Formula: see text] be factorial domains containing [Formula: see text]. In this paper, we give a criterion, in terms of locally nilpotent derivations, for [Formula: see text] to be [Formula: see text]-isomorphic to [Formula: see text], where [Formula: see text] is nonzero and [Formula: see text]. As a consequence, we retrieve a recent result due to Masuda [Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc. 145(4) (2017) 1439–1452] characterizing Danielewski hypersurfaces whose coordinate ring is factorial. We also apply our criterion to the study of triangular

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11

Ferrero, Miguel, Yves Lequain, and Andrzej Nowicki. "A note on locally nilpotent derivations." Journal of Pure and Applied Algebra 79, no. 1 (1992): 45–50. http://dx.doi.org/10.1016/0022-4049(92)90125-y.

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12

Pakovich, F. B. "Locally nilpotent derivations of polynomial rings." Mathematical Notes 58, no. 2 (1995): 890–91. http://dx.doi.org/10.1007/bf02304113.

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13

El Kahoui, M’hammed, Najoua Essamaoui та Mustapha Ouali. "A nilpotency criterion for derivations over reduced ℚ-algebras". International Journal of Algebra and Computation 31, № 05 (2021): 903–13. http://dx.doi.org/10.1142/s0218196721500429.

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Let [Formula: see text] be a reduced ring containing [Formula: see text] and let [Formula: see text] be commuting locally nilpotent derivations of [Formula: see text]. In this paper, we give an algorithm to decide the local nilpotency of derivations of the form [Formula: see text], where [Formula: see text] are elements in [Formula: see text].

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14

Nielsen, Pace P., and Michał Ziembowski. "Derivations and bounded nilpotence index." International Journal of Algebra and Computation 25, no. 03 (2015): 433–38. http://dx.doi.org/10.1142/s0218196715500034.

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We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x;δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about locally nilpotent PI rings.

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15

Gupta, Neena, and Sourav Sen. "Locally nilpotent derivations of double Danielewski surfaces." Journal of Pure and Applied Algebra 224, no. 4 (2020): 106208. http://dx.doi.org/10.1016/j.jpaa.2019.106208.

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16

Barkatou, Moulay A., Hassan El Houari, and M’hammed El Kahoui. "Triangulable locally nilpotent derivations in dimension three." Journal of Pure and Applied Algebra 212, no. 9 (2008): 2129–39. http://dx.doi.org/10.1016/j.jpaa.2008.01.001.

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17

Bergen, Jeffrey, and Piotr Grzeszczuk. "On Rings with Locally Nilpotent Skew Derivations." Communications in Algebra 39, no. 10 (2011): 3698–708. http://dx.doi.org/10.1080/00927872.2010.510816.

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18

Daowsud, Katthaleeya, Monrudee Sirivoravit, Utsanee Leerawat, and Nigel Byott. "On locally nilpotent derivations of Boolean semirings." Cogent Mathematics 4, no. 1 (2017): 1351064. http://dx.doi.org/10.1080/23311835.2017.1351064.

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19

van den Essen, A. "Locally nilpotent derivations and their applications, III." Journal of Pure and Applied Algebra 98, no. 1-2 (1995): 15–23. http://dx.doi.org/10.1016/0022-4049(94)00036-i.

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20

Daigle, D. "On some properties of locally nilpotent derivations." Journal of Pure and Applied Algebra 114, no. 3 (1997): 221–30. http://dx.doi.org/10.1016/0022-4049(95)00173-5.

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21

van den Essen, Arno. "Locally nilpotent derivations and their applications, III." Journal of Pure and Applied Algebra 98, no. 1 (1995): 15–23. http://dx.doi.org/10.1016/0022-4049(95)90013-6.

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22

Bergen, Jeffrey, and Piotr Grzeszczuk. "Locally nilpotent skew derivations with central invariants." Journal of Algebra 483 (August 2017): 71–82. http://dx.doi.org/10.1016/j.jalgebra.2017.03.040.

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23

Daigle, Daniel, and Gene Freudenburg. "Locally Nilpotent Derivations over a UFD and an Application to Rank Two Locally Nilpotent Derivations ofk[X1,…,Xn]." Journal of Algebra 204, no. 2 (1998): 353–71. http://dx.doi.org/10.1006/jabr.1998.7465.

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24

Gaifullin, Sergey, and Yulia Zaitseva. "On homogeneous locally nilpotent derivations of trinomial algebras." Journal of Algebra and Its Applications 18, no. 10 (2019): 1950196. http://dx.doi.org/10.1142/s0219498819501962.

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We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots of trinomial algebras.

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25

Daigle, Daniel. "Locally nilpotent derivations and the structure of rings." Journal of Pure and Applied Algebra 224, no. 4 (2020): 106201. http://dx.doi.org/10.1016/j.jpaa.2019.106201.

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26

Freudenburg, Gene, and Lucy Moser-Jauslin. "Locally nilpotent derivations of rings with roots adjoined." Michigan Mathematical Journal 62, no. 2 (2013): 227–58. http://dx.doi.org/10.1307/mmj/1370870372.

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27

Hadasf, Ofer, and Leonid Makar-Limanov. "Newton polytopes of constants of locally nilpotent derivations." Communications in Algebra 28, no. 8 (2000): 3667–78. http://dx.doi.org/10.1080/00927870008827048.

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28

Zaitseva, Yu I. "Homogeneous Locally Nilpotent Derivations of Nonfactorial Trinomial Algebras." Mathematical Notes 105, no. 5-6 (2019): 818–30. http://dx.doi.org/10.1134/s0001434619050201.

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29

El Kahoui, M'hammed. "UFDs with commuting linearly independent locally nilpotent derivations." Journal of Algebra 289, no. 2 (2005): 446–52. http://dx.doi.org/10.1016/j.jalgebra.2005.01.048.

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30

Kuroda, Shigeru. "Fields defined by locally nilpotent derivations and monomials." Journal of Algebra 293, no. 2 (2005): 395–406. http://dx.doi.org/10.1016/j.jalgebra.2005.06.011.

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31

BERGEN, JEFFREY, and PIOTR GRZESZCZUK. "GOLDIE DIMENSION OF CONSTANTS OF LOCALLY NILPOTENT SKEW DERIVATIONS." Journal of Algebra and Its Applications 11, no. 06 (2012): 1250105. http://dx.doi.org/10.1142/s0219498812501058.

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In this paper, we examine rings R with locally nilpotent skew derivations d and compare the Goldie dimension of R to that of the subring of constants Rd. This generalizes the situation where one compares the Goldie dimension of an Ore extension to that of the base ring. Under certain natural conditions placed upon Rd, we show that R and Rd have the same Goldie dimension.

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32

Daigle, D. "Homogeneous locally nilpotent derivations of k[X, Y, Z]." Journal of Pure and Applied Algebra 128, no. 2 (1998): 109–32. http://dx.doi.org/10.1016/s0022-4049(97)00043-1.

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33

LIU, DAYAN, and XIAOSONG SUN. "THE FACTORIAL CONJECTURE AND IMAGES OF LOCALLY NILPOTENT DERIVATIONS." Bulletin of the Australian Mathematical Society 101, no. 1 (2019): 71–79. http://dx.doi.org/10.1017/s0004972719000546.

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The factorial conjecture was proposed by van den Essen et al. [‘On the image conjecture’, J. Algebra 340(1) (2011), 211–224] to study the image conjecture, which arose from the Jacobian conjecture. We show that the factorial conjecture holds for all homogeneous polynomials in two variables. We also give a variation of the result and use it to show that the image of any linear locally nilpotent derivation of $\mathbb{C}[x,y,z]$ is a Mathieu–Zhao subspace.

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34

Barkatou, Moulay A., and M’hammed El Kahoui. "Locally nilpotent derivations with a PID ring of constants." Proceedings of the American Mathematical Society 140, no. 1 (2011): 119–28. http://dx.doi.org/10.1090/s0002-9939-2011-10962-6.

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35

Bergen, Jeffrey, and Piotr Grzeszczuk. "Gelfand-Kirillov dimension of algebras with locally nilpotent derivations." Israel Journal of Mathematics 206, no. 1 (2015): 313–25. http://dx.doi.org/10.1007/s11856-015-1152-1.

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36

El Kahoui, Mʼhammed, and Mustapha Ouali. "Fixed point free locally nilpotent derivations of A2-fibrations." Journal of Algebra 372 (December 2012): 480–87. http://dx.doi.org/10.1016/j.jalgebra.2012.09.025.

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37

Bhatwadekar, S. M., and Amartya K. Dutta. "Kernel of locally nilpotent $R$-derivations of $R[X,Y]$." Transactions of the American Mathematical Society 349, no. 8 (1997): 3303–19. http://dx.doi.org/10.1090/s0002-9947-97-01946-6.

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38

Bianchi, Angelo Calil, and Marcelo Oliveira Veloso. "Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces." Journal of Algebra 469 (January 2017): 96–108. http://dx.doi.org/10.1016/j.jalgebra.2016.08.030.

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39

Ferrero, Miguel, Antonio Giambruno, and César Polcino Milies. "A Note on Derivations of Group Rings." Canadian Mathematical Bulletin 38, no. 4 (1995): 434–37. http://dx.doi.org/10.4153/cmb-1995-063-8.

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AbstractLetRGdenote the group ring of a groupGover a semiprime ringR. We prove that, if the center ofGis of finite index and some natural restrictions hold, then everyR-derivation ofRGis inner. We also give an example of a groupGwhich is both locally finite and nilpotent and such that, for every fieldF, there exists anF-derivation ofFGwhich is not inner.

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40

Lopes, Samuel A. "Non-Noetherian generalized Heisenberg algebras." Journal of Algebra and Its Applications 16, no. 04 (2017): 1750064. http://dx.doi.org/10.1142/s0219498817500645.

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In this note, we classify the non-Noetherian generalized Heisenberg algebras [Formula: see text] introduced in [R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291]. In case deg [Formula: see text] > 1, we determine all locally finite and also all locally nilpotent derivations of [Formula: see text] and describe the automorphism group of these algebras.

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41

Essen, Arno Van Den. "Locally Finite and Locally Nilpotent Derivations with Applications to Polynomial Flows and Polynomial Morphism." Proceedings of the American Mathematical Society 116, no. 3 (1992): 861. http://dx.doi.org/10.2307/2159458.

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42

van den Essen, Arno. "Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms." Proceedings of the American Mathematical Society 116, no. 3 (1992): 861. http://dx.doi.org/10.1090/s0002-9939-1992-1111440-5.

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43

Romaskevich, Elena. "Sums and commutators of homogeneous locally nilpotent derivations of fiber type." Journal of Pure and Applied Algebra 218, no. 3 (2014): 448–55. http://dx.doi.org/10.1016/j.jpaa.2013.06.014.

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44

Bodnarchuk, Yu V., and P. H. Prokof’ev. "Locally nilpotent derivations and Nagata-type utomorphisms of a polynomial algebra." Ukrainian Mathematical Journal 61, no. 8 (2009): 1199–214. http://dx.doi.org/10.1007/s11253-010-0271-4.

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45

Sun, Xiaosong, and Dayan Liu. "Images of locally nilpotent derivations of polynomial algebras in three variables." Journal of Algebra 569 (March 2021): 401–15. http://dx.doi.org/10.1016/j.jalgebra.2020.10.025.

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46

Liendo, Alvaro. "Affine $ \mathbb{T} $ -varieties of complexity one and locally nilpotent derivations." Transformation Groups 15, no. 2 (2010): 389–425. http://dx.doi.org/10.1007/s00031-010-9089-2.

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47

Daigle, Daniel. "On polynomials in three variables annihilated by two locally nilpotent derivations." Journal of Algebra 310, no. 1 (2007): 303–24. http://dx.doi.org/10.1016/j.jalgebra.2006.12.005.

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48

Masuda, Kayo. "Homogeneous locally nilpotent derivations having slices and embeddings of affine spaces." Journal of Algebra 321, no. 6 (2009): 1719–33. http://dx.doi.org/10.1016/j.jalgebra.2008.12.012.

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49

Babu, Janaki Raman, and Prosenjit Das. "Structure of A2-fibrations having fixed point free locally nilpotent derivations." Journal of Pure and Applied Algebra 225, no. 12 (2021): 106763. http://dx.doi.org/10.1016/j.jpaa.2021.106763.

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50

Dangovski, Rumen, Vesselin Drensky, and Şehmus Fındık. "Weitzenböck derivations of free metabelian associative algebras." Journal of Algebra and Its Applications 16, no. 03 (2017): 1750041. http://dx.doi.org/10.1142/s0219498817500414.

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By the classical theorem of Weitzenböck the algebra of constants [Formula: see text] of a nonzero locally nilpotent linear derivation [Formula: see text] of the polynomial algebra [Formula: see text] in several variables over a field [Formula: see text] of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants [Formula: see text] of a locally nilpotent linear derivation [Formula: see text] of a finitely generated relatively free algebra [Formula: see text] in a variety [Formula: see text] of unitary associative algebras over [Formula:

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