Characterization of irreducible polynomials over a special principal ideal ring

Brahim Boudine

Displaying similar documents to “Characterization of irreducible polynomials over a special principal ideal ring”

On near-ring ideals with $(σ, τ)$ -derivation

Öznur Golbaşi, Neşet Aydin (2007)

Archivum Mathematicum

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Let $N$ be a $3$ -prime left near-ring with multiplicative center $Z$ , a $(σ, τ)$ -derivation $D$ on $N$ is defined to be an additive endomorphism satisfying the product rule $D (x y) = τ (x) D (y) + D (x) σ (y)$ for all $x, y \in N$ , where $σ$ and $τ$ are automorphisms of $N$ . A nonempty subset $U$ of $N$ will be called a semigroup right ideal (resp. semigroup left ideal) if $U N \subset U$ (resp. $N U \subset U$ ) and if $U$ is both a semigroup right ideal and a semigroup left ideal, it be called a semigroup ideal. We prove the following results: Let $D$ be a $(σ,$ $τ)$ -derivation...

C(X) vs. C(X) modulo its socle

F. Azarpanah, O. A. S. Karamzadeh, S. Rahmati (2008)

Colloquium Mathematicae

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Let $C_{F} (X)$ be the socle of C(X). It is shown that each prime ideal in $C (X) / C_{F} (X)$ is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that $d i m (C (X) / C_{F} (X)) \geq d i m C (X)$ , where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points....

Normal restrictions of the noncofinal ideal on $P_{κ} (λ)$

Pierre Matet (2013)

Fundamenta Mathematicae

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We discuss the problem of whether there exists a restriction of the noncofinal ideal on $P_{κ} (λ)$ that is normal.

On the uniform behaviour of the Frobenius closures of ideals

K. Khashyarmanesh (2007)

Colloquium Mathematicae

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Let be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q() be the smallest positive integer m such that ${(^{F})}^{[p^{m}]} =^{[p^{m}]}$ , where $^{F}$ is the Frobenius closure of . This paper is concerned with the question whether the set $Q (^{[p^{m}]}) : m \in ℕ ₀$ is bounded. We give an affirmative answer in the case that the ideal is generated by an u.s.d-sequence c₁,..., cₙ for R such that (i) the map $R / \sum_{j = 1}^{n} R c_{j} \to R / \sum_{j = 1}^{n} R c ²_{j}$ induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all $\in a s s (c ₁^{j}, . . ., c ₙ^{j})$ , c₁/1,..., cₙ/1 is a $R_{}$ -filter...

The Briançon-Skoda number of analytic irreducible planar curves

Jacob Sznajdman (2014)

Annales de l’institut Fourier

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The Briançon-Skoda number of a ring $R$ is defined as the smallest integer k, such that for any ideal $I \subset R$ and $l \geq 1$ , the integral closure of $I^{k + l - 1}$ is contained in $I^{l}$ . We compute the Briançon-Skoda number of the local ring of any analytic irreducible planar curve in terms of its Puiseux characteristics. It turns out that this number is closely related to the Milnor number.

Skew inverse power series rings over a ring with projective socle

Kamal Paykan (2017)

Czechoslovak Mathematical Journal

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A ring $R$ is called a right $PS$ -ring if its socle, $Soc (R_{R})$ , is projective. Nicholson and Watters have shown that if $R$ is a right $PS$ -ring, then so are the polynomial ring $R [x]$ and power series ring $R [[x]]$ . In this paper, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does the skew inverse power series ring $R [[x^{- 1}; α, δ]]$ and the skew polynomial ring $R [x; α, δ]$ , where $R$ is an associative ring equipped with an automorphism $α$ and an $α$ -derivation $δ$ . Our results extend and unify many existing...

On wsq-primary ideals

Emel Aslankarayiğit Uğurlu, El Mehdi Bouba, Ünsal Tekir, Suat Koç (2023)

Czechoslovak Mathematical Journal

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We introduce weakly strongly quasi-primary (briefly, wsq-primary) ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity and $Q$ a proper ideal of $R$ . The proper ideal $Q$ is said to be a weakly strongly quasi-primary ideal if whenever $0 \neq a b \in Q$ for some $a, b \in R$ , then $a^{2} \in Q$ or $b \in \sqrt{Q} .$ Many examples and properties of wsq-primary ideals are given. Also, we characterize nonlocal Noetherian von Neumann regular rings, fields, nonlocal rings over which every proper ideal is wsq-primary, and zero...

More on the strongly 1-absorbing primary ideals of commutative rings

Ali Yassine, Mohammad Javad Nikmehr, Reza Nikandish (2024)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with identity. We study the concept of strongly 1-absorbing primary ideals which is a generalization of $n$ -ideals and a subclass of $1$ -absorbing primary ideals. A proper ideal $I$ of $R$ is called strongly 1-absorbing primary if for all nonunit elements $a, b, c \in R$ such that $a b c \in I$ , it is either $a b \in I$ or $c \in \sqrt{0}$ . Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings $R$ over which every semi-primary ideal is strongly 1-absorbing primary, and rings $R$ over which...

A generalisation of Amitsur's A-polynomials

Adam Owen, Susanne Pumplün (2021)

Communications in Mathematics

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We find examples of polynomials $f \in D [t; σ, δ]$ whose eigenring $ℰ (f)$ is a central simple algebra over the field $F = C \cap Fix (σ) \cap Const (δ)$ .

On skew derivations as homomorphisms or anti-homomorphisms

Mohd Arif Raza, Nadeem ur Rehman, Shuliang Huang (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$ . In this manuscript, we investigate the action of skew derivation $(δ, ϕ)$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$ . Moreover, we provide an example for semiprime case.

On the Anderson-Badawi $ω_{R [X]} (I [X]) = ω_{R} (I)$ conjecture

Peyman Nasehpour (2016)

Archivum Mathematicum

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Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$ -absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$ -absorbing ideal of $R$ , if whenever $x_{1} \dots x_{n + 1} \in I$ for $x_{1}, ..., x_{n + 1} \in R$ , then there are $n$ of the $x_{i}$ ’s whose product is in $I$ and conjecture that $ω_{R [X]} (I [X]) = ω_{R} (I)$ for any ideal $I$ of an arbitrary ring $R$ , where $ω_{R} (I) = min {n : I is an n -absorbing ideal of R}$ . In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following...

M-ideals of homogeneous polynomials

Verónica Dimant (2011)

Studia Mathematica

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We study the problem of whether $_{w} (ⁿ E)$ , the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space (ⁿE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if $_{w} (ⁿ E)$ is an M-ideal in (ⁿE), then $_{w} (ⁿ E)$ coincides with $_{w 0} (ⁿ E)$ (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if $_{w} (ⁿ E) =_{w 0} (ⁿ E)$ and (E) is an...