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

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

Journal of the European Mathematical Society

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

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...

### A commutativity theorem for associative rings

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 ${x}^{p}\left[{x}^{n},y\right]{x}^{q}={x}^{r}\left[x,{y}^{m}\right]{y}^{s}$ or ${x}^{p}\left[{x}^{n},y\right]{x}^{q}={y}^{s}\left[x,{y}^{m}\right]{x}^{r}$ 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 $x,y\in R,m\left[x,y\right]=0$ implies $\left[x,y\right]=0$).

### Rings of $\left(\gamma ,\delta \right)$ type

Portugaliae mathematica

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Algebra i Logika

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### Commutativity of rings through a Streb’s result

Czechoslovak Mathematical Journal

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In this paper we investigate commutativity of rings with unity satisfying any one of the properties: $\begin{array}{cc}& \left\{1-g\left(y{x}^{m}\right)\right\}\phantom{\rule{4pt}{0ex}}\left[y{x}^{m}-{x}^{r}f\left(y{x}^{m}\right)\phantom{\rule{4pt}{0ex}}{x}^{s},x\right]\left\{1-h\left(y{x}^{m}\right)\right\}=0,\hfill \\ & \left\{1-g\left(y{x}^{m}\right)\right\}\phantom{\rule{4pt}{0ex}}\left[{x}^{m}y-{x}^{r}f\left(y{x}^{m}\right){x}^{s},x\right]\left\{1-h\left(y{x}^{m}\right)\right\}=0,\hfill \\ & {y}^{t}\left[x,{y}^{n}\right]=g\left(x\right)\left[f\left(x\right),y\right]h\left(x\right)\phantom{\rule{4pt}{0ex}}\mathrm{a}nd\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left[x,{y}^{n}\right]\phantom{\rule{4pt}{0ex}}{y}^{t}=g\left(x\right)\left[f\left(x\right),y\right]h\left(x\right)\hfill \end{array}$ for some $f\left(X\right)$ in ${X}^{2}ℤ\left[X\right]$ and $g\left(X\right)$, $h\left(X\right)$ in $ℤ\left[X\right]$, 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...

### Left APP-property of formal power series rings

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 $R\left[\left[x;\alpha \right]\right]$ 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 $R\left[\left[x;\alpha \right]\right]$ is left APP if and only if for any sequence $\left({b}_{0},{b}_{1},\cdots \right)$ 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...

### Vortex rings for the Gross-Pitaevskii equation

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 $N=3$).

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

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,...