A unified approach to the Armendariz property of polynomial rings and power series rings

Tsiu-Kwen Lee; Yiqiang Zhou

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

Skeletons, bodies and generalized $E (R)$ -algebras

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2009)

Journal of the European Mathematical Society

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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]: = $\sum_{i \geq 0} r_{i} x^{i} \in R [[x]]$ : ∃ 0 ≤ n∈ ℤ such that $r_{i} \in I$ , ∀ 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 $m > 1, s \geq 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p = p (x) \geq 0, q = q (x) \geq 0, n = n (x) \geq 0, r = r (x) \geq 0$ 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 $y \in R$ . In the present paper it is shown that $R$ is commutative if it satisfies the property $Q (m)$ (i.e. for all $x, y \in R, m [x, y] = 0$ implies $[x, y] = 0$ ).

Rings of $(γ, δ)$ type

Kleinfeld, Erwin (1959)

Portugaliae mathematica

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On a theorem of McCoy

Rajendra K. Sharma, Amit B. Singh (2024)

Mathematica Bohemica

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We study McCoy’s theorem to the skew Hurwitz series ring $(HR, ω)$ for some different classes of rings such as: semiprime rings, APP rings and skew Hurwitz serieswise quasi-Armendariz rings. Moreover, we establish an equivalence relationship between a right zip ring and its skew Hurwitz series ring in case when a ring $R$ satisfies McCoy’s theorem of skew Hurwitz series.

О регулярных самоинъективных и о $V$ -кольцах

Д.В. Тюкавкин, D. V. Tjukavkin, D. V. Tǔkavkin, D. V. Tjukavkin (1994)

Algebra i Logika

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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: $\begin{matrix} {1 - g (y x^{m})} [y x^{m} - x^{r} f (y x^{m}) x^{s}, x] {1 - h (y x^{m})} = 0, \\ {1 - g (y x^{m})} [x^{m} y - x^{r} f (y x^{m}) x^{s}, x] {1 - h (y x^{m})} = 0, \\ y^{t} [x, y^{n}] = g (x) [f (x), y] h (x) a n d [x, y^{n}] y^{t} = g (x) [f (x), y] h (x) \end{matrix}$ for some $f (X)$ in $X^{2} ℤ [X]$ and $g (X)$ , $h (X)$ in $ℤ [X]$ , where $m \geq 0$ , $r \geq 0$ , $s \geq 0$ , $n > 0$ , $t > 0$ 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 $a \in R$ . 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 $(b_{0}, b_{1}, \dots)$ of elements of $R$ the ideal $l_{R}$ $(\sum_{j = 0}^{\infty} \sum_{k = 0}^{\infty} R α^{k} (b_{j}))$ 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 $N \geq 3$ . 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 $N = 3$ ).