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”

Rings with divisibility on descending chains of ideals

Oussama Aymane Es Safi, Najib Mahdou, Ünsal Tekir (2024)

Czechoslovak Mathematical Journal

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This paper deals with the rings which satisfy $D C C_{d}$ condition. This notion has been introduced recently by R. Dastanpour and A. Ghorbani (2017) as a generalization of Artnian rings. It is of interest to investigate more deeply this class of rings. This study focuses on commutative case. In this vein, we present this work in which we examine the transfer of these rings to the trivial, amalgamation and polynomial ring extensions. We also investigate the relationship between this class of 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...

Note on strongly nil clean elements in rings

Aleksandra Kostić, Zoran Z. Petrović, Zoran S. Pucanović, Maja Roslavcev (2019)

Czechoslovak Mathematical Journal

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Let $R$ be an associative unital ring and let $a \in R$ be a strongly nil clean element. We introduce a new idea for examining the properties of these elements. This approach allows us to generalize some results on nil clean and strongly nil clean rings. Also, using this technique many previous proofs can be significantly shortened. Some shorter proofs concerning nil clean elements in rings in general, and in matrix rings in particular, are presented, together with some generalizations of these...

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...