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.

Differential polynomial rings in several variables over locally nilpotent rings

2019 ◽  
Vol 30 (01) ◽  
pp. 117-123 ◽  
Author(s):  
Fei Yu Chen ◽  
Hannah Hagan ◽  
Allison Wang
Keyword(s):  
Polynomial Ring ◽  
Differential Polynomial ◽  
Polynomial Rings ◽  
Nilpotent Ring ◽  
Several Variables ◽  
Locally Nilpotent ◽  
Differential Polynomial Ring

We show that a differential polynomial ring over a locally nilpotent ring in several commuting variables is Behrens radical, extending a result by Chebotar.

Derivations and bounded nilpotence index

2015 ◽  
Vol 25 (03) ◽  
pp. 433-438 ◽  
Author(s):  
Pace P. Nielsen ◽  
Michał Ziembowski
Keyword(s):  
Polynomial Ring ◽  
Differential Polynomial ◽  
Prime Radical ◽  
Polynomial Rings ◽  
Locally Nilpotent ◽  
Differential Polynomial Ring ◽  
Index Of Nilpotence ◽  
Nil Ring ◽  
Strong Barrier

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.

scholarly journals MAXIMAL COMMUTATIVE SUBRINGS AND SIMPLICITY OF ORE EXTENSIONS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250192 ◽  
Author(s):  
JOHAN ÖINERT ◽  
JOHAN RICHTER ◽  
SERGEI D. SILVESTROV
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Commutative Subring

The aim of this paper is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x; id R, δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x; σ, δ] is a maximal commutative subring containing R and, in the case when σ = id R, we show that it intersects every nonzero ideal of R[x; id R, δ] nontrivially. Using this we show that if R is δ-simple and maximal commutative in R[x; id R, δ], then R[x; id R, δ] is simple. We also show that under some conditions on R the converse holds.

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.

scholarly journals Quasi-duo differential polynomial rings

2018 ◽  
Vol 17 (04) ◽  
pp. 1850072 ◽  
Author(s):  
Mai Hoang Bien ◽  
Johan Öinert
Keyword(s):  
Polynomial Ring ◽  
Differential Polynomial ◽  
Polynomial Rings ◽  
Partial Answer ◽  
Maximal Ideals ◽  
Differential Polynomial Ring

In this paper, we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide nontrivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.

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 $\mathbb{Q}$.

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.

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