### Skeletons, bodies and generalized $E\left(R\right)$-algebras

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2009)

Journal of the European Mathematical Society

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Rüdiger Göbel, Daniel Herden, Saharon Shelah (2009)

Journal of the European Mathematical Society

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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]: = ${\sum}_{i\ge 0}{r}_{i}{x}^{i}\in R\left[\left[x\right]\right]$ : ∃ 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...

Mohammad Ashraf (1995)

Archivum Mathematicum

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Let $m>1,s\ge 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p=p\left(x\right)\ge 0,q=q\left(x\right)\ge 0,n=n\left(x\right)\ge 0,r=r\left(x\right)\ge 0$ 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 $Q\left(m\right)$ (i.e. for all $x,y\in R,m[x,y]=0$ implies $[x,y]=0$).

Kleinfeld, Erwin (1959)

Portugaliae mathematica

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

Algebra i Logika

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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: $$\begin{array}{cc}& \{1-g\left(y{x}^{m}\right)\}\phantom{\rule{4pt}{0ex}}[y{x}^{m}-{x}^{r}f\left(y{x}^{m}\right)\phantom{\rule{4pt}{0ex}}{x}^{s},x]\{1-h\left(y{x}^{m}\right)\}=0,\hfill \\ & \{1-g\left(y{x}^{m}\right)\}\phantom{\rule{4pt}{0ex}}[{x}^{m}y-{x}^{r}f\left(y{x}^{m}\right){x}^{s},x]\{1-h\left(y{x}^{m}\right)\}=0,\hfill \\ & {y}^{t}[x,{y}^{n}]=g\left(x\right)[f\left(x\right),y]h\left(x\right)\phantom{\rule{4pt}{0ex}}\mathrm{a}nd\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}[x,{y}^{n}]\phantom{\rule{4pt}{0ex}}{y}^{t}=g\left(x\right)[f\left(x\right),y]h\left(x\right)\hfill \end{array}$$ for some $f\left(X\right)$ in ${X}^{2}\mathbb{Z}\left[X\right]$ and $g\left(X\right)$, $h\left(X\right)$ in $\mathbb{Z}\left[X\right]$, where $m\ge 0$, $r\ge 0$, $s\ge 0$, $n>0$, $t>0$ 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...

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 ${l}_{R}\left(Ra\right)$ 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 $R\left[\right[x;\alpha \left]\right]$ where $\alpha $ is a ring automorphism of $R$. It is shown that if $R$ is a ring satisfying descending chain condition on right annihilators then $R\left[\right[x;\alpha \left]\right]$ is left APP if and only if for any sequence $({b}_{0},{b}_{1},\cdots )$ of elements of $R$ the ideal ${l}_{R}$ $\left({\sum}_{j=0}^{\infty}{\sum}_{k=0}^{\infty}R{\alpha}^{k}\left({b}_{j}\right)\right)$ is right $s$-unital. As an application we give a sufficient condition under which...

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 $N\ge 3$. 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 $N=3$).

Vincent Thilliez (2002)

Studia Mathematica

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In rings ${\Gamma}_{M}$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M={\left({M}_{l}\right)}_{l\ge 0}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in ${\Gamma}_{M}$ 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 ${\Gamma}_{M}$, provided ${\Gamma}_{M}$ is stable under derivation. By a further development of the method, we obtain the main result of the paper,...