Displaying similar documents to “A unified approach to the Armendariz property of polynomial rings and power series rings”

An intermediate ring between a polynomial ring and a power series ring

M. Tamer Koşan, Tsiu-Kwen Lee, Yiqiang Zhou (2013)

Colloquium Mathematicae

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Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R;I][x] the following subring of R[[x]]: [R;I][x]: = i 0 r i x i R [ [ x ] ] : ∃ 0 ≤ n∈ ℤ such that r i I , ∀ i ≥ n. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor R[[x]] is isomorphic to [R;I][x]. The results characterizing polynomial rings or power series rings with...

A commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

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Let m > 1 , s 1 be fixed positive integers, and let R be a ring with unity 1 in which for every x in R there exist integers p = p ( x ) 0 , q = q ( x ) 0 , n = n ( x ) 0 , r = r ( x ) 0 such that either x p [ x n , y ] x q = x r [ x , y m ] y s or x p [ x n , y ] x q = y s [ x , y m ] x r for all y R . In the present paper it is shown that R is commutative if it satisfies the property Q ( m ) (i.e. for all x , y R , m [ x , y ] = 0 implies [ x , y ] = 0 ).

Commutativity of rings through a Streb’s result

Moharram A. Khan (2000)

Czechoslovak Mathematical Journal

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In this paper we investigate commutativity of rings with unity satisfying any one of the properties: { 1 - g ( y x m ) } [ y x m - x r f ( y x m ) x s , x ] { 1 - h ( y x m ) } = 0 , { 1 - g ( y x m ) } [ x m y - x r f ( y x m ) x s , x ] { 1 - h ( y x m ) } = 0 , y t [ x , y n ] = g ( x ) [ f ( x ) , y ] h ( x ) a n d [ x , y n ] y t = g ( x ) [ f ( x ) , y ] h ( x ) for some f ( X ) in X 2 [ X ] and g ( X ) , h ( X ) in [ X ] , where m 0 , r 0 , s 0 , n > 0 , t > 0 are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements x and y for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results...

Left APP-property of formal power series rings

Zhongkui Liu, Xiao Yan Yang (2008)

Archivum Mathematicum

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A ring R is called a left APP-ring if the left annihilator l R ( R a ) is right s -unital as an ideal of R for any element a R . We consider left APP-property of the skew formal power series ring R [ [ x ; α ] ] where α is a ring automorphism of R . It is shown that if R is a ring satisfying descending chain condition on right annihilators then R [ [ x ; α ] ] is left APP if and only if for any sequence ( b 0 , b 1 , ) of elements of R the ideal l R ( j = 0 k = 0 R α k ( b j ) ) is right s -unital. As an application we give a sufficient condition under which...

Vortex rings for the Gross-Pitaevskii equation

Fabrice Bethuel, G. Orlandi, Didier Smets (2004)

Journal of the European Mathematical Society

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We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension N 3 . We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if N = 3 ).

Bounds for quotients in rings of formal power series with growth constraints

Vincent Thilliez (2002)

Studia Mathematica

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In rings Γ M of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence M = ( M l ) l 0 (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in Γ M such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to Γ M , provided Γ M is stable under derivation. By a further development of the method, we obtain the main result of the paper,...