On wsq-primary ideals

Emel Aslankarayiğit Uğurlu; El Mehdi Bouba; Ünsal Tekir; Suat Koç

Displaying similar documents to “On wsq-primary ideals”

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...

On quasi $n$ -ideals of commutative rings

Adam Anebri, Najib Mahdou, Emel Aslankarayiğit Uğurlu (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of $n$ -ideals and the class of $(2, n)$ -ideals. A proper ideal $I$ of $R$ is said to be a quasi $n$ -ideal if $\sqrt{I}$ is an $n$ -ideal of $R .$ Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the $n$ -ideals, the quasi primary ideals, the $(2, n)$ -ideals and the $p r$ -ideals. Moreover, we use the quasi $n$ -ideals to characterize...

On norm closed ideals in $L (ℓ_{p}, ℓ_{q})$

B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V. G. Troitsky (2007)

Studia Mathematica

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It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X = ℓ_{p}$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L (ℓ_{p} \oplus ℓ_{q})$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L (ℓ_{p}, ℓ_{q})$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L (ℓ_{p}, ℓ_{q})$ , including one that has not been studied before. The proofs use various methods...

Monomial ideals with tiny squares and Freiman ideals

Ibrahim Al-Ayyoub, Mehrdad Nasernejad (2021)

Czechoslovak Mathematical Journal

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We provide a construction of monomial ideals in $R = K [x, y]$ such that $μ (I^{2}) < μ (I)$ , where $μ$ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring $R$ , we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on $μ (I^{k})$ that generalize...

$α$ -ideals in $0$ -distributive posets

Khalid A. Mokbel (2015)

Mathematica Bohemica

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The concept of $α$ -ideals in posets is introduced. Several properties of $α$ -ideals in $0$ -distributive posets are studied. Characterization of prime ideals to be $α$ -ideals in $0$ -distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal $I$ of a $0$ -distributive poset is non-dense, then $I$ is an $α$ -ideal. Moreover, it is shown that the set of all $α$ -ideals $α Id (P)$ of a poset $P$ with $0$ forms a complete lattice. A result analogous to separation theorem for...

$(δ, 2)$ -primary ideals of a commutative ring

Gülşen Ulucak, Ece Yetkin Çelikel (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with nonzero identity, let $ℐ (ℛ)$ be the set of all ideals of $R$ and $δ : ℐ (ℛ) \to ℐ (ℛ)$ an expansion of ideals of $R$ defined by $I \mapsto δ (I)$ . We introduce the concept of $(δ, 2)$ -primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(δ, 2)$ -primary ideal if whenever $a, b \in R$ and $a b \in I$ , then $a^{2} \in I$ or $b^{2} \in δ (I)$ . Our purpose is to extend the concept of $2$ -ideals to $(δ, 2)$ -primary ideals of commutative rings. Then we investigate the basic properties of $(δ, 2)$ -primary ideals and also discuss the relations among $(δ, 2)$ -primary, $δ$ -primary...

$0$ -ideals in $0$ -distributive posets

Khalid A. Mokbel (2016)

Mathematica Bohemica

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The concept of a $0$ -ideal in $0$ -distributive posets is introduced. Several properties of $0$ -ideals in $0$ -distributive posets are established. Further, the interrelationships between $0$ -ideals and $α$ -ideals in $0$ -distributive posets are investigated. Moreover, a characterization of prime ideals to be $0$ -ideals in $0$ -distributive posets is obtained in terms of non-dense ideals. It is shown that every $0$ -ideal of a $0$ -distributive meet semilattice is semiprime. Several counterexamples are discussed. ...

Semiproper ideals

Hiroshi Sakai (2005)

Fundamenta Mathematicae

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We say that an ideal I on $_{κ} λ$ is semiproper if the corresponding poset $ℙ_{I}$ is semiproper. In this paper we investigate properties of semiproper ideals on $_{κ} λ$ .

The strong persistence property and symbolic strong persistence property

Mehrdad Nasernejad, Kazem Khashyarmanesh, Leslie G. Roberts, Jonathan Toledo (2022)

Czechoslovak Mathematical Journal

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Let $I$ be an ideal in a commutative Noetherian ring $R$ . Then the ideal $I$ has the strong persistence property if and only if $(I^{k + 1} :_{R} I) = I^{k}$ for all $k$ , and $I$ has the symbolic strong persistence property if and only if $(I^{(k + 1)} :_{R} I^{(1)}) = I^{(k)}$ for all $k$ , where $I^{(k)}$ denotes the $k$ th symbolic power of $I$ . We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial...

Semi $n$ -ideals of commutative rings

Ece Yetkin Çelikel, Hani A. Khashan (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with identity. A proper ideal $I$ is said to be an $n$ -ideal of $R$ if for $a, b \in R$ , $a b \in I$ and $a \notin \sqrt{0}$ imply $b \in I$ . We give a new generalization of the concept of $n$ -ideals by defining a proper ideal $I$ of $R$ to be a semi $n$ -ideal if whenever $a \in R$ is such that $a^{2} \in I$ , then $a \in \sqrt{0}$ or $a \in I$ . We give some examples of semi $n$ -ideal and investigate semi $n$ -ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new...

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....

When spectra of lattices of $z$ -ideals are Stone-Čech compactifications

Themba Dube (2017)

Mathematica Bohemica

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Let $X$ be a completely regular Hausdorff space and, as usual, let $C (X)$ denote the ring of real-valued continuous functions on $X$ . The lattice of $z$ -ideals of $C (X)$ has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) $β X$ precisely when $X$ is a $P$ -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a $d$ -ideal if whenever two elements have the same annihilator...

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.