On relative pure cyclic fields with power integral bases

Mohammed Sahmoudi; Mohammed Elhassani Charkani

Displaying similar documents to “On relative pure cyclic fields with power integral bases”

Augmentation quotients for Burnside rings of generalized dihedral groups

Shan Chang (2016)

Czechoslovak Mathematical Journal

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Let $H$ be a finite abelian group of odd order, $𝒟$ be its generalized dihedral group, i.e., the semidirect product of $C_{2}$ acting on $H$ by inverting elements, where $C_{2}$ is the cyclic group of order two. Let $Ω (𝒟)$ be the Burnside ring of $𝒟$ , $Δ (𝒟)$ be the augmentation ideal of $Ω (𝒟)$ . Denote by $Δ^{n} (𝒟)$ and $Q_{n} (𝒟)$ the $n$ th power of $Δ (𝒟)$ and the $n$ th consecutive quotient group $Δ^{n} (𝒟) / Δ^{n + 1} (𝒟)$ , respectively. This paper provides an explicit $ℤ$ -basis for $Δ^{n} (𝒟)$ and determines the isomorphism class of $Q_{n} (𝒟)$ for each positive integer $n$ .

(Generalized) filter properties of the amalgamated algebra

Yusof Azimi (2022)

Archivum Mathematicum

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Let $R$ and $S$ be commutative rings with unity, $f : R \to S$ a ring homomorphism and $J$ an ideal of $S$ . Then the subring $R ⋈^{f} J : = {(a, f (a) + j) ∣ a \in R$ and $j \in J}$ of $R \times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$ . In this paper, we determine when $R ⋈^{f} J$ is a (generalized) filter ring.

Maximal non $λ$ -subrings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with unity. The notion of maximal non $λ$ -subrings is introduced and studied. A ring $R$ is called a maximal non $λ$ -subring of a ring $T$ if $R \subset T$ is not a $λ$ -extension, and for any ring $S$ such that $R \subset S \subseteq T$ , $S \subseteq T$ is a $λ$ -extension. We show that a maximal non $λ$ -subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D + M$ construction is a maximal non $λ$ -domain is given. A necessary condition...

The clean elements of the ring $ℛ (L)$

Ali Akbar Estaji, Maryam Taha (2024)

Czechoslovak Mathematical Journal

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We characterize clean elements of $ℛ (L)$ and show that $α \in ℛ (L)$ is clean if and only if there exists a clopen sublocale $U$ in $L$ such that $𝔠_{L} (coz (α - 1)) \subseteq U \subseteq 𝔬_{L} (coz (α))$ . Also, we prove that $ℛ (L)$ is clean if and only if $ℛ (L)$ has a clean prime ideal. Then, according to the results about $ℛ (L),$ we immediately get results about $𝒞_{c} (L) .$

On monogenity of certain pure number fields of degrees $2^{r} \cdot 3^{k} \cdot 7^{s}$

Hamid Ben Yakkou, Jalal Didi (2024)

Mathematica Bohemica

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Let $K = ℚ (α)$ be a pure number field generated by a complex root $α$ of a monic irreducible polynomial $F (x) = x^{2^{r} \cdot 3^{k} \cdot 7^{s}} - m \in ℤ [x]$ , where $r$ , $k$ , $s$ are three positive natural integers. The purpose of this paper is to study the monogenity of $K$ . Our results are illustrated by some examples.

On prolongations of rank one discrete valuations

Lhoussain El Fadil (2019)

Commentationes Mathematicae Universitatis Carolinae

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Let $(K, ν)$ be a valued field, where $ν$ is a rank one discrete valuation. Let $R$ be its ring of valuation, $𝔪$ its maximal ideal, and $L$ an extension of $K$ , defined by a monic irreducible polynomial $F (X) \in R [X]$ . Assume that $\bar{F} (X)$ factors as a product of $r$ distinct powers of monic irreducible polynomials. In this paper a condition which guarantees the existence of exactly $r$ distinct valuations of $K$ extending $ν$ is given, in such a way that it generalizes the results given in the paper “Prolongations of valuations...

Equations in the Hadamard ring of rational functions

Andrea Ferretti, Umberto Zannier (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $K$ be a number field. It is well known that the set of recurrencesequences with entries in $K$ is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume ${a_{n}}$ is a recurrence sequence and suppose that all the $a_{n}$ have a $d^{th}$ root in the field...

Annihilators of skew derivations with Engel conditions on prime rings

Taylan Pehlivan, Emine Albas (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring $Q$ , $C$ the extended centroid of $R$ and $a \in R$ . Suppose that $δ$ is a nonzero $σ$ -derivation of $R$ such that $a {[δ (x^{n}), x^{n}]}_{k} = 0$ for all $x \in R$ , where $σ$ is an automorphism of $R$ , $n$ and $k$ are fixed positive integers. Then $a = 0$ .

On minimal ideals in the ring of real-valued continuous functions on a frame

Abolghasem Karimi Feizabadi, Ali Akbar Estaji, Mostafa Abedi (2018)

Archivum Mathematicum

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Let $ℛ L$ be the ring of real-valued continuous functions on a frame $L$ . The aim of this paper is to study the relation between minimality of ideals $I$ of $ℛ L$ and the set of all zero sets in $L$ determined by elements of $I$ . To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame $L$ , it is proved that the $f$ -ring $ℛ L$ is isomorphic to the $f$ -ring $C (Σ L)$ of all real continuous functions on the topological space $Σ L$ . Finally, a one-one correspondence...

On extending $C^{k}$ functions from an open set to $ℝ$ with applications

Walter D. Burgess, Robert M. Raphael (2023)

Czechoslovak Mathematical Journal

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For $k \in ℕ \cup {\infty}$ and $U$ open in $ℝ$ , let $C^{k} (U)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown that for $f \in C^{k} (U)$ there is $g \in C^{\infty} (ℝ)$ with $U \subseteq coz g$ and $h \in C^{k} (ℝ)$ with ${f g |}_{U} {= h |}_{U}$ . The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$ . The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring $C^{k} (ℝ)$ are deduced from this, in particular that $Q_{cl} (C^{k} (ℝ)) = Q (C^{k} (ℝ))$ .

Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings

Vincenzo de Filippis (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_{r}$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n \geq 1$ a fixed positive integer. Let $α$ be an automorphism of the ring $R$ . An additive map $D : R \to R$ is called an $α$ -derivation (or a skew derivation) on $R$ if $D (x y) = D (x) y + α (x) D (y)$ for all $x, y \in R$ . An additive mapping $F : R \to R$ is called a generalized $α$ -derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F (x y) = F (x) y + α (x) D (y)$ for all $x, y \in R$ . We prove...