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.

On α-nilpotent elements and α-Armendariz rings

2015 ◽  
Vol 14 (05) ◽  
pp. 1550064
Author(s):  
Hong Kee Kim ◽  
Nam Kyun Kim ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Hidetoshi Marubayashi
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
Armendariz Ring

Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced the concept of nil-Armendariz property as a generalization. Hong et al. studied Armendariz property on skew polynomial rings and introduced the notion of an α-Armendariz ring, where α is a ring monomorphism. In this paper, we investigate the structure of the set of α-nilpotent elements in α-Armendariz rings and introduce an α-nil-Armendariz ring. We examine the set of [Formula: see text]-nilpotent elements in a skew polynomial ring R[x;α], where [Formula: see text] is the monomorphism induced by the monomorphism α of an α-Armendariz ring R. We prove that every polynomial with α-nilpotent coefficients in a ring R is [Formula: see text]-nilpotent when R is of bounded index of α-nilpotency, and moreover, R is shown to be α-nil-Armendariz in this situation. We also characterize the structure of the set of α-nilpotent elements in α-nil-Armendariz rings, and investigate the relations between α-(nil-)Armendariz property and other standard ring theoretic properties.

scholarly journals A Characterization of separable polynomials over a skew polynomial ring

1985 ◽  
Vol 38 (2) ◽  
pp. 275-280
Author(s):  
George Szeto
Keyword(s):  
Commutative Ring ◽  
Finite Order ◽  
Polynomial Ring ◽  
Subject Classification ◽  
Skew Polynomial Ring ◽  
Skew Polynomial

AbstractThe characterization of a separable polynomial over an indecomposable commutative ring (with no idempotents but 0 and 1) in terms of the discriminant proved by G. J. Janusz is generalized to a skew polynomial ring R [ X, ρ] over a not necessarily commutative ring R where ρ is an automorphism of R with a finite order. 1980 Mathematics subject classification (Amer. Math. Soc.): 16 A 05.

Nilpotent Elements and Skew Polynomial Rings

2012 ◽  
Vol 19 (spec01) ◽  
pp. 821-840 ◽  
Author(s):  
A. Alhevaz ◽  
A. Moussavi ◽  
E. Hashemi
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Special Cases ◽  
Ring Extensions ◽  
Skew Polynomial

We study the structure of the set of nilpotent elements in extended semicommutative rings and introduce nil α-semicommutative rings as a generalization. We resolve the structure of nil α-semicommutative rings and obtain various necessary or sufficient conditions for a ring to be nil α-semicommutative, unifying and generalizing a number of known commutative-like conditions in special cases. We also classify which of the standard nilpotence properties on polynomial rings pass to skew polynomial ring. Constructing various examples, we classify how the nil α-semicommutative rings behaves under various ring extensions. Also, we consider the nil-Armendariz condition on a skew polynomial ring.

COMPLETELY PRIME ONE-SIDED IDEALS IN SKEW POLYNOMIAL RINGS

2021 ◽  
pp. 1-8
Author(s):  
GIL ALON ◽  
ELAD PARAN
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Skew Polynomial Ring ◽  
Skew Polynomial Rings ◽  
Skew Polynomial ◽  
Prime Elements

Abstract Let R = K[x, σ] be the skew polynomial ring over a field K, where σ is an automorphism of K of finite order. We show that prime elements in R correspond to completely prime one-sided ideals – a notion introduced by Reyes in 2010. This extends the natural correspondence between prime elements and prime ideals in commutative polynomial rings.

Extended Centroids of Skew Polynomial Rings

1985 ◽  
Vol 28 (1) ◽  
pp. 67-76 ◽  
Author(s):  
Jerry D. Rosen ◽  
Mary Peles Rosen
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Prime Ring ◽  
Polynomial Rings ◽  
Extended Centroid ◽  
Skew Polynomial Ring ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

AbstractLet R be a prime ring with σ ∊ Aut (R). We determine the extended centroid of the skew polynomial ring R[x, σ] when (i) 〈σ〉 is X-outer of finite order, (ii) 〈σ〉 is X-outer and infinite, (iii) σm is X-inner and no smaller power of σ fixes the extended centroid of R.

Jacobson Radicals of Skew Polynomial Rings of Derivation Type

2014 ◽  
Vol 57 (3) ◽  
pp. 609-613 ◽  
Author(s):  
Alireza Nasr-Isfahani
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomial Rings ◽  
Base Ring ◽  
Necessary And Sufficient ◽  
Skew Polynomial

AbstractWe provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

ON WEAK ZIP SKEW POLYNOMIAL RINGS

2012 ◽  
Vol 05 (03) ◽  
pp. 1250039
Author(s):  
R. Mohammadi ◽  
A. Moussavi ◽  
M. Zahiri
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
General Situation ◽  
Skew Polynomial Ring ◽  
Glasgow Math ◽  
Skew Polynomial Rings ◽  
Ore Extensions ◽  
Skew Polynomial ◽  
J 51

We introduce the notion of nil(α, δ)-compatible rings which is a generalization of reduced rings and (α, δ)-compatible rings. In [Ore extensions of weak zip rings, Glasgow Math. J.51 (2009) 525–537] Ouyang introduces the notion of right (respectively, left) weak zip rings and proved that, a ring R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip, when R is (α, δ)-compatible and reversible. We extend this result to the more general situation that, when R has (α, δ)-condition and quasi-IFP, then nil (R)[x; α, δ] = nil (R[x; α, δ]); and R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip.

SKEW POLYNOMIAL RINGS WHICH ARE GENERALIZED ASANO PRIME RINGS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350024 ◽  
Author(s):  
H. MARUBAYASHI ◽  
INTAN MUCHTADI-ALAMSYAH ◽  
A. UEDA
Keyword(s):  
Polynomial Ring ◽  
Prime Ring ◽  
Quotient Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Prime Rings ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

Let R be a prime Goldie ring with quotient ring Q and σ be an automorphism of R. We define (σ-) generalized Asano prime rings and prove that a skew polynomial ring R[x; σ] is a generalized Asano prime ring if and only if R is a σ-generalized Asano prime ring. This is done by giving explicitly the structure of all v-ideals of R[x; σ] in case R is a σ-Krull prime ring. We provide some examples of σ-generalized Asano prime rings which are not Krull prime rings.

Isomorphisms between skew polynomial rings

1978 ◽  
Vol 25 (3) ◽  
pp. 314-321 ◽  
Author(s):  
M. Rimmer
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

AbstractLet R be any ring and α any automorphism of R. We determine here all the automorphisms of the skew polynomial ring R[x, α] which fix R elementwise. We then deal with isomorphisms between different skew polynomial rings whose underlying rings are isomorphic.

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