Strongly fixed ideals in $ C (L)$ and compact frames

A. A. Estaji; A. Karimi Feizabadi; M. Abedi

Displaying similar documents to “Strongly fixed ideals in $C (L)$ and compact frames”

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 $_{κ} λ$ .

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...

Two ideals connected with strong right upper porosity at a point

Viktoriia Bilet, Oleksiy Dovgoshey, Jürgen Prestin (2015)

Czechoslovak Mathematical Journal

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Let $SP$ be the set of upper strongly porous at $0$ subsets of $ℝ^{+}$ and let $\hat{I} (SP)$ be the intersection of maximal ideals $I \subseteq SP$ . Some characteristic properties of sets $E \in \hat{I} (SP)$ are obtained. We also find a characteristic property of the intersection of all maximal ideals contained in a given set which is closed under subsets. It is shown that the ideal generated by the so-called completely strongly porous at $0$ subsets of $ℝ^{+}$ is a proper subideal of $\hat{I} (SP) .$ Earlier, completely strongly porous sets and some of their properties...

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...

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...

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...

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...

$(δ, 2)$ -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...

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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If $A$ is a domain with the ascending chain condition on (integral) invertible ideals, then the group $I(A)$ of its invertible ideals is generated by the set $I_{m}(A)$ of maximal invertible ideals. In this note we study some properties of $I_{m}(A)$ and we prove that, if $I(A)$ is a free group on $I_{m}(A)$ , then $A$ is a locally factorial Krull domain.

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...

Weak precompactness and property (V*) in spaces of compact operators

Ioana Ghenciu (2015)

Colloquium Mathematicae

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We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w *} (E *, F)$ (resp. $K_{w *} (E *, F)$ ) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w *} (E *, F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w *} (E *, F)$ has property (wV*). Suppose...

The group ring $𝕂 F$ of Richard Thompson’s Group $F$ has no minimal non-zero ideals

John Donnelly (2019)

Archivum Mathematicum

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We use a total order on Thompson’s group $F$ to show that the group ring $𝕂 F$ has no minimal non-zero ideals.

Effective Nullstellensatz for arbitrary ideals

János Kollár (1999)

Journal of the European Mathematical Society

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Let $f_{i}$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials $g_{i}$ such that $\sum g_{i} f_{i} = 1$ . The effective versions of this result bound the degrees of the $g_{i}$ in terms of the degrees of the $f_{j}$ . The aim of this paper is to generalize this to the case when the $f_{i}$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.

Weakly null sequences with upper estimates

Daniel Freeman (2008)

Studia Mathematica

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We prove that if $(v_{i})$ is a seminormalized basic sequence and X is a Banach space such that every normalized weakly null sequence in X has a subsequence that is dominated by $(v_{i})$ , then there exists a uniform constant C ≥ 1 such that every normalized weakly null sequence in X has a subsequence that is C-dominated by $(v_{i})$ . This extends a result of Knaust and Odell, who proved this for the cases in which $(v_{i})$ is the standard basis for $ℓ_{p}$ or c₀.

Generalization of the $S$ -Noetherian concept

Abdelamir Dabbabi, Ali Benhissi (2023)

Archivum Mathematicum

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Let $A$ be a commutative ring and $𝒮$ a multiplicative system of ideals. We say that $A$ is $𝒮$ -Noetherian, if for each ideal $Q$ of $A$ , there exist $I \in 𝒮$ and a finitely generated ideal $F \subseteq Q$ such that $I Q \subseteq F$ . In this paper, we study the transfer of this property to the polynomial ring and Nagata’s idealization.

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...

Displaying similar documents to “Strongly fixed ideals in C ( L ) and compact frames”

Displaying similar documents to “Strongly fixed ideals in $C (L)$ and compact frames”