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 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.

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.

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.

Simplicity of iterated Ore extensions

2021 ◽  
pp. 2250052
Author(s):  
Mamta Balodi ◽  
Sumit Kumar Upadhyay
Keyword(s):  
Polynomial Ring ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Commutative Domain ◽  
Necessary And Sufficient ◽  
Commutative Subring

Here we study the simplicity of an iterated Ore extension of a unital ring [Formula: see text]. We give necessary conditions for the simplicity of an iterated Ore extension when [Formula: see text] is a commutative domain. A class of iterated Ore extensions, namely the differential polynomial ring [Formula: see text] in [Formula: see text]-variables is considered. The conditions for a commutative domain [Formula: see text] of characteristic zero to be a maximal commutative subring of its differential polynomial ring [Formula: see text] are given, and the necessary and sufficient conditions for [Formula: see text] to be simple are also found.

scholarly journals Nilpotents and units in skew polynomial rings over commutative rings

1979 ◽  
Vol 28 (4) ◽  
pp. 423-426 ◽  
Author(s):  
M. Rimmer ◽  
K. R. Pearson
Keyword(s):  
Commutative Ring ◽  
Finite Order ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

AbstractLet R be a commutative ring with an automorphism ∞ of finite order n. An element f of the skew polynomial ring R[x, α] is nilpotent if and only if all coefficients of fn are nilpotent. (The case n = 1 is the well-known description of the nilpotent elements of the ordinary polynomial ring R[x].) A characterization of the units in R[x, α] is also given.

Automorphisms of a Certain Skew Polynomial Ring of Derivation Type

1990 ◽  
Vol 42 (6) ◽  
pp. 949-958
Author(s):  
Isao Kikumasa
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Prime Integer ◽  
Ring Homomorphisms ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

Throughout this paper, all rings have the identity 1 and ring homomorphisms are assumed to preserve 1. We use p to denote a prime integer and F to denote a field of characteristic p. For an element α in F, we setA = F[ϰ]/(ϰp - α)F[ϰ].Moreover, by D and R, we denote the derivation of A induced by the ordinary derivation of F[ϰ] and the skew polynomial ring A[X,D] where aX = Xa+D(a) (a ∈ A), respectively (cf. [2]).In [3], R. W. Gilmer determined all the B-automorphisms of B[X] for any commutative ring B. Since then, some extensions or generalizations of his results have been obtained ([1], [2] and [5]). As to the characterization of automorphisms of skew polynomial rings, M. Rimmer [5] established a thorough result in case of automorphism type, while M. Ferrero and K. Kishimoto [2], among others, have made some progress in case of derivation type.

scholarly journals Characterization of Some Fully Ordered Rings

1971 ◽  
Vol 12 (1) ◽  
pp. 83-85
Author(s):  
M. Satyanarayana
Keyword(s):  
Polynomial Ring ◽  
Positive Element ◽  
Polynomial Rings ◽  
Large Element ◽  
Ring Of Integers ◽  
Bounded Set ◽  
Lexicographic Ordering

In a fully ordered (f.o.) ring with identity, the set of all bounded elements as defined below might be an Archimedian subring. Most of the examples of f.o. rings constructed in literature having the bounded set as Archimedian subring are polynomial rings. For example I[x], R[x] etc., where I is the ring of integers and R is the field of rationals, with lexicographic ordering. Now we ask whether a f.o. ring with identity, with the set of bounded elements as Archimedian subring can be a polynomial ring over an Archimedian subring. This is answered affirmatively in Theorem 1. It is proved in Theorem 3 that f.o. rings with identity and with every positive element a large element, belong to the above class. The problem then arises as to when the set of all bounded elements, called a weak Archimedian sub- ring in [2], becomes an Archimedian subring. This problem is completely solved in Theorem 2. The concept of weak Archimedian rings is found to be useful by the author in characterizing some f.o. rings as algebraic algebras in [3].

A GENERALIZATION OF REDUCED RINGS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250070 ◽  
Author(s):  
A. R. NASR-ISFAHANI ◽  
A. MOUSSAVI
Keyword(s):  
Polynomial Ring ◽  
Differential Polynomial ◽  
Prime Ideals ◽  
Armendariz Rings ◽  
Differential Polynomial Ring ◽  
Minimal Prime ◽  
Armendariz Ring ◽  
Ring Derivation

For a ring derivation δ, we introduce and investigate a generalization of reduced rings and Armendariz rings which we call a δ-Armendariz ring. Various classes of δ-Armendariz rings is provided and a number of properties of this generalization are established. Radicals and minimal prime ideals of the differential polynomial ring R[x; δ], in terms of those of a δ-Armendariz R, is determined. We prove that several properties transfer between R and the differential polynomial ring R[x; δ], in case R is δ-Armendariz.

scholarly journals Prime and maximal ideals in polynomial rings

1995 ◽  
Vol 37 (3) ◽  
pp. 351-362 ◽  
Author(s):  
Miguel Ferrero
Keyword(s):  
Polynomial Ring ◽  
Maximal Ideal ◽  
Identity Element ◽  
Minimal Degree ◽  
Polynomial Rings ◽  
Maximal Ideals ◽  
Equivalent Conditions

In this paper we study prime and maximal ideals in a polynomial ring R[X], where R is a ring with identity element. It is well-known that to study many questions we may assume Ris prime and consider just R-disjoint ideals. We give a characterizaton for an R-disjoint ideal to be prime. We study conditions under which there exists an R-disjoint ideal which is a maximal ideal and when this is the case how to determine all such maximal ideals. Finally, we prove a theorem giving several equivalent conditions for a maximal ideal to be generated by polynomials of minimal degree.

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