Interpolation in the Automorphism Group of a Polynomial Ring

2020 ◽  
Vol 27 (03) ◽  
pp. 587-598
Author(s):  
M’hammed El Kahoui ◽  
Najoua Essamaoui ◽  
Mustapha Ouali
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Proper Ideal ◽  
Polynomial Rings ◽  
Group Homomorphism ◽  
Volume Preserving

Let R be a commutative ring with unity and SAn(R) be the group of volume-preserving automorphisms of the polynomial R-algebra R[n]. Given a proper ideal 𝔞 of R, we address in this paper the question of whether the canonical group homomorphism SAn(R) → SAn(R/𝔞) is surjective. As an application, we retrieve and generalize, in a unified way, several known results on residual coordinates in polynomial rings.

scholarly journals Unique factorization in polynomial rings with zero divisors

2020 ◽  
pp. 2150113 ◽  
Author(s):  
D. D. Anderson ◽  
Ranthony A. C. Edmonds
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Unique Factorization ◽  
Factorization Property ◽  
Zero Divisors ◽  
Unique Factorization Domain

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

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.

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
Author(s):  
Unsal Tekir ◽  
Suat Koc ◽  
Kursat Oral
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

scholarly journals The structure of the automorphism group of polynomial rings

1973 ◽  
Vol 25 (1) ◽  
pp. 25-39 ◽  
Author(s):  
Marshall Fraser ◽  
Adolf Mader
Keyword(s):  
Automorphism Group ◽  
Polynomial Rings

Affine Actions of Uq(sl(2)) on Polynomial Rings

2015 ◽  
Vol 58 (2) ◽  
pp. 233-240
Author(s):  
Jeffrey Bergen
Keyword(s):  
Polynomial Ring ◽  
Scalar Multiplication ◽  
Polynomial Rings ◽  
Affine Actions

AbstractWe classify the affine actions of Uq(sl(2)) on commutative polynomial rings in m ≥ 1 variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either m or m−1 variables, depending upon whether q is or is not a root of 1.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

scholarly journals Combinatorial Reductions for the Stanley Depth of $I$ and $S/I$

10.37236/6783 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Mitchel T. Keller ◽  
Stephen J. Young
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Strict Inequality ◽  
Polynomial Rings ◽  
Computer Search ◽  
Stanley Depth ◽  
The Relationship

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal $I$ and the Stanley depth of its compliment, $S/I$. Using these results we are able to prove that if $S$ is a polynomial ring with at most 5 indeterminates and $I$ is a square-free monomial ideal, then the Stanley depth of $S/I$ is strictly larger than the Stanley depth of $I$. Using a computer search, we are able to extend this strict inequality up to polynomial rings with at most 7 indeterminates. This partially answers questions asked by Propescu and Qureshi as well as Herzog.

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Proper Ideal ◽  
Graph Structure ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

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.

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