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Relevant bibliographies by topics / Locally Nilpotent Derivation

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Journal articles

Academic literature on the topic 'Locally Nilpotent Derivation'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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Journal articles on the topic "Locally Nilpotent Derivation"

1

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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2

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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3

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 of [Formula: see text] is a Mathieu subspace [W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra 214 (2010) 1200–1216; Mathieu subspaces of associative algebras, J. Algebra 350(2) (2012) 245–272] of [Formula: see text], and whether or not a locally nilpotent (LN) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] maps every ideal of [Formula: see text] to a Mathieu subspace of [Formula: see text]. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring [Formula: see text] is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for LF or LN algebraic derivations and [Formula: see text]-[Formula: see text]-derivations of integral domains of characteristic zero.

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4

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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5

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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6

Freudenburg, Gene. "A note on the kernel of a locally nilpotent derivation." Proceedings of the American Mathematical Society 124, no. 1 (1996): 27–29. http://dx.doi.org/10.1090/s0002-9939-96-03003-1.

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7

BERGEN, JEFFREY, and PIOTR GRZESZCZUK. "SKEW POWER SERIES RINGS OF DERIVATION TYPE." Journal of Algebra and Its Applications 10, no. 06 (2011): 1383–99. http://dx.doi.org/10.1142/s0219498811005245.

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In this paper, we contrast the structure of a noncommutative algebra R with that of the skew power series ring R[[y;d]]. Several of our main results examine when the rings R, Rd, and R[[y;d]] are prime or semiprime under the assumption that d is a locally nilpotent derivation.

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8

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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9

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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10

Arzhantsev, Ivan. "On rigidity of factorial trinomial hypersurfaces." International Journal of Algebra and Computation 26, no. 05 (2016): 1061–70. http://dx.doi.org/10.1142/s0218196716500442.

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An affine algebraic variety [Formula: see text] is rigid if the algebra of regular functions [Formula: see text] admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least [Formula: see text].

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