Some notes on the composite $G$-valuations

Angeliki Kontolatou

Displaying similar documents to “Some notes on the composite $G$ -valuations”

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...

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

Abdelhafed Elkhadiri (2011)

Annales Polonici Mathematici

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Let R be a real closed field, and denote by $_{R, n}$ the ring of germs, at the origin of Rⁿ, of $C^{\infty}$ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring $_{R, n} \subset_{R, n}$ with some natural properties. We prove that, for each n ∈ ℕ, $_{R, n}$ is a noetherian ring and if R = ℝ (the field of real numbers), then $_{ℝ, n} = ₙ$ , where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring $_{R, n}$ . ...

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...

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...

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...

(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.

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 $n$ .

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.

About G-rings

Najib Mahdou (2017)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if $R \subseteq T$ is a ring extension such that $m T \subseteq R$ for some regular element $m$ of $T$ , then $T$ is a G-ring if and only if so is $R$ . Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.

Displaying similar documents to “Some notes on the composite G -valuations”

Displaying similar documents to “Some notes on the composite $G$ -valuations”