On some noetherian rings of $C^{∞}$ germs on a real closed field

Abdelhafed Elkhadiri

Displaying similar documents to “On some noetherian rings of $C^{\infty}$ germs on a real closed field”

On weakened $(α, δ)$ -skew Armendariz rings

Alireza Majdabadi Farahani, Mohammad Maghasedi, Farideh Heydari, Hamidagha Tavallaee (2022)

Mathematica Bohemica

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In this note, for a ring endomorphism $α$ and an $α$ -derivation $δ$ of a ring $R$ , the notion of weakened $(α, δ)$ -skew Armendariz rings is introduced as a generalization of $α$ -rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(α, δ)$ -skew Armendariz ring if and only if $T_{n} (R)$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz if and only if $R [x] / (x^{n})$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz ring for any 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.

P-injective group rings

Liang Shen (2020)

Czechoslovak Mathematical Journal

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A ring $R$ is called right P-injective if every homomorphism from a principal right ideal of $R$ to $R_{R}$ can be extended to a homomorphism from $R_{R}$ to $R_{R}$ . Let $R$ be a ring and $G$ a group. Based on a result of Nicholson and Yousif, we prove that the group ring $RG$ is right P-injective if and only if (a) $R$ is right P-injective; (b) $G$ is locally finite; and (c) for any finite subgroup $H$ of $G$ and any principal right ideal $I$ of $RH$ , if $f \in {Hom}_{R} (I_{R}, R_{R})$ , then there exists $g \in {Hom}_{R} ({RH}_{R}, R_{R})$ such that ${g |}_{I} = f$ . Similarly, we also obtain equivalent...

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...

Symmetric and reversible properties of bi-amalgamated rings

Antonysamy Aruldoss, Chelliah Selvaraj (2024)

Czechoslovak Mathematical Journal

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Let $f : A \to B$ and $g : A \to C$ be two ring homomorphisms and let $K$ and $K^{'}$ be two ideals of $B$ and $C$ , respectively, such that $f^{- 1} (K) = g^{- 1} (K^{'})$ . We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring $A ⋈^{f, g} (K, K^{'})$ of $A$ with $(B, C)$ along $(K, K^{'})$ with respect to $(f, g)$ .

A note on Skolem-Noether algebras

Juncheol Han, Tsiu-Kwen Lee, Sangwon Park (2021)

Czechoslovak Mathematical Journal

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The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let $K$ be a unital commutative ring, not necessarily a field. Given a unital $K$ -algebra $S$ , where $K$ is contained in the center of $S$ , $n \in ℕ$ , the goal of this paper is to study the question: when can a homomorphism $φ : M_{n} (K) \to M_{n} (S)$ be extended to an inner automorphism of $M_{n} (S)$ ? As an application of main results presented in the paper, it is proved that if $S$ is...

On relative pure cyclic fields with power integral bases

Mohammed Sahmoudi, Mohammed Elhassani Charkani (2023)

Mathematica Bohemica

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Let $L = K (α)$ be an extension of a number field $K$ , where $α$ satisfies the monic irreducible polynomial $P (X) = X^{p} - β$ of prime degree belonging to $𝔬_{K} [X]$ ( $𝔬_{K}$ is the ring of integers of $K$ ). The purpose of this paper is to study the monogenity of $L$ over $K$ by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field...

Strongly 2-nil-clean rings with involutions

Huanyin Chen, Marjan Sheibani Abdolyousefi (2019)

Czechoslovak Mathematical Journal

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A $*$ -ring $R$ is strongly 2-nil- $*$ -clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$ -rings are obtained. We prove that a $*$ -ring $R$ is strongly 2-nil- $*$ -clean if and only if for all $a \in R$ , $a^{2} \in R$ is strongly nil- $*$ -clean, if and only if for any $a \in R$ there exists a $*$ -tripotent $e \in R$ such that $a - e \in R$ is nilpotent and $e a = a e$ , if and only if $R$ is a strongly $*$ -clean SN ring, if and only if $R$ is abelian, $J (R)$ is nil and $R / J (R)$ is $*$ -tripotent. Furthermore, we explore...

Notes on generalizations of Bézout rings

Haitham El Alaoui, Hakima Mouanis (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we give new characterizations of the $P$ - $2$ -Bézout property of trivial ring extensions. Also, we investigate the transfer of this property to homomorphic images and to finite direct products. Our results generate original examples which enrich the current literature with new examples of non- $2$ -Bézout $P$ - $2$ -Bézout rings and examples of non- $P$ -Bézout $P$ - $2$ -Bézout rings.

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...

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...

Semicommutativity of the rings relative to prime radical

Handan Kose, Burcu Ungor (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$ -semicommutative. We prove that a ring $R$ is $P$ -semicommutative if and only if $R [x]$ is $P$ -semicommutative if and only if $R [x, x^{- 1}]$ is $P$ -semicommutative. Also, if $R [[x]]$ is $P$ -semicommutative, then $R$ is $P$ -semicommutative. The converse holds provided that $P (R)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$ , $R$ is $P$ -semicommutative...

Displaying similar documents to “On some noetherian rings of C ∞ germs on a real closed field”

Displaying similar documents to “On some noetherian rings of $C^{\infty}$ germs on a real closed field”