scholarly journals ON SEPARABLE AND -FORMS

2018 ◽  
Vol 239 ◽  
pp. 346-354
Author(s):  
AMARTYA KUMAR DUTTA ◽  
NEENA GUPTA ◽  
ANIMESH LAHIRI
Keyword(s):  
Characteristic Zero ◽  
One Dimensional ◽  
Noetherian Domain ◽  
Locally Nilpotent Derivation ◽  
Locally Nilpotent ◽  
Nilpotent Derivation

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.

Pascal finite polynomial automorphisms

2019 ◽  
Vol 18 (07) ◽  
pp. 1950124
Author(s):  
Elżbieta Adamus ◽  
Paweł Bogdan ◽  
Teresa Crespo ◽  
Zbigniew Hajto
Keyword(s):  
Characteristic Zero ◽  
Positive Characteristic ◽  
Exponential Map ◽  
Locally Finite ◽  
Arbitrary Characteristic ◽  
Polynomial Automorphisms ◽  
Locally Nilpotent Derivation ◽  
Locally Nilpotent ◽  
Nilpotent Derivation

The class of Pascal finite polynomial automorphisms is a subclass of the class of locally finite ones allowing a more effective approach. In characteristic zero, a Pascal finite automorphism is the exponential map of a locally nilpotent derivation. However, Pascal finite automorphisms are defined in any characteristic, and therefore constitute a generalization of exponential automorphisms to positive characteristic. In this paper, we prove several properties of Pascal finite automorphisms. We obtain in particular that the Pascal finite property is stable under taking powers but not under composition. This leads us to formulate a generalization of the exponential generators conjecture to arbitrary characteristic.

Cartan Subalgebras of

2003 ◽  
Vol 46 (4) ◽  
pp. 597-616 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Ivan Penkov
Keyword(s):  
Finite Rank ◽  
Characteristic Zero ◽  
Cartan Subalgebra ◽  
Adjoint Representation ◽  
Linear Functionals ◽  
Maximal Subalgebra ◽  
Locally Finite ◽  
One Dimensional ◽  
Cartan Subalgebras ◽  
Locally Nilpotent

AbstractLet V be a vector space over a field of characteristic zero and V* be a space of linear functionals on V which separate the points of V. We consider V ⊗ V* as a Lie algebra of finite rank operators on V, and set (V, V*) := V ⊗ V*. We define a Cartan subalgebra of (V, V*) as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of (V;V*) under the assumption that is algebraically closed. A subalgebra of (V, V*) is a Cartan subalgebra if and only if it equals for some one-dimensional subspaces Vj ⊆ V and (Vj)* ⊆ V* with (Vi)* (Vj) = δij and such that the spaces . We then discuss explicit constructions of subspaces Vj and (Vj)* as above. Our second main result claims that a Cartan subalgebra of (V, V*) can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra h which coincides with the maximal locally nilpotent h-submodule of (V, V*), and such that the adjoint representation of is locally finite.

THE FACTORIAL CONJECTURE AND IMAGES OF LOCALLY NILPOTENT DERIVATIONS

2019 ◽  
Vol 101 (1) ◽  
pp. 71-79 ◽  
Author(s):  
DAYAN LIU ◽  
XIAOSONG SUN
Keyword(s):  
Jacobian Conjecture ◽  
Homogeneous Polynomials ◽  
Locally Nilpotent Derivation ◽  
Locally Nilpotent ◽  
Nilpotent Derivation

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.

SKEW POWER SERIES RINGS OF DERIVATION TYPE

2011 ◽  
Vol 10 (06) ◽  
pp. 1383-1399 ◽  
Author(s):  
JEFFREY BERGEN ◽  
PIOTR GRZESZCZUK
Keyword(s):  
Power Series ◽  
Power Series Ring ◽  
Noncommutative Algebra ◽  
Series Ring ◽  
Locally Nilpotent Derivation ◽  
Locally Nilpotent ◽  
Nilpotent Derivation ◽  
Power Series Rings

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.

scholarly journals On rigidity of factorial trinomial hypersurfaces

2016 ◽  
Vol 26 (05) ◽  
pp. 1061-1070 ◽  
Author(s):  
Ivan Arzhantsev
Keyword(s):  
Algebraic Variety ◽  
Regular Functions ◽  
Locally Nilpotent Derivation ◽  
Locally Nilpotent ◽  
Nilpotent Derivation

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].

ON THE CONNECTION BETWEEN DIFFERENTIAL POLYNOMIAL RINGS AND POLYNOMIAL RINGS OVER NIL RINGS

2019 ◽  
Vol 101 (3) ◽  
pp. 438-441
Author(s):  
LOUISA CATALANO ◽  
MEGAN CHANG-LEE
Keyword(s):  
Polynomial Ring ◽  
Positive Characteristic ◽  
Differential Polynomial ◽  
Polynomial Rings ◽  
Locally Nilpotent Derivation ◽  
Locally Nilpotent ◽  
Nilpotent Derivation ◽  
Differential Polynomial Ring ◽  
Field Of Positive Characteristic ◽  
Algebra Over A Field

In this paper, we study some connections between the polynomial ring $R[y]$ and the differential polynomial ring $R[x;D]$. In particular, we answer a question posed by Smoktunowicz, which asks whether $R[y]$ is nil when $R[x;D]$ is nil, provided that $R$ is an algebra over a field of positive characteristic and $D$ is a locally nilpotent derivation.

scholarly journals $$v_c$$-Noetherian domains and Krull domains

2021 ◽  
Author(s):  
Gyu Whan Chang
Keyword(s):  
Dedekind Domain ◽  
Closed Domain ◽  
One Dimensional ◽  
Star Operation ◽  
Krull Domain ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Krull Domains

AbstractLet D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

Sign in / Sign up

Export Citation Format

Share Document