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

F. Azarpanah; O. A. S. Karamzadeh; S. Rahmati

Displaying similar documents to “C(X) vs. C(X) modulo its socle”

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...

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...

Ideals in big Lipschitz algebras of analytic functions

Thomas Vils Pedersen (2004)

Studia Mathematica

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For 0 < γ ≤ 1, let $Λ ⁺_{γ}$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{γ} (E, Q)$ be the closed ideal in $Λ ⁺_{γ}$ consisting of those functions $f \in Λ ⁺_{γ}$ for which (i) f = 0 on E, (ii) $| f (z) - f (w) | = o (| z - w |^{γ})$ as d(z,E),d(w,E) → 0, (iii) $f / Q \in Λ ⁺_{γ}$ . Also, for a closed ideal I in $Λ ⁺_{γ}$ , let $E_{I}$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_{I}$ be the greatest common divisor of the inner parts of non-zero functions...

Duality of measures of non-𝒜-compactness

Juan Manuel Delgado, Cándido Piñeiro (2015)

Studia Mathematica

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Let be a Banach operator ideal. Based on the notion of -compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non–compactness of an operator. We consider a map $χ_{}$ (respectively, $n_{}$ ) acting on the operators of the surjective (respectively, injective) hull of such that $χ_{} (T) = 0$ (respectively, $n_{} (T) = 0$ ) if and only if the operator T is -compact (respectively, injectively -compact). Under certain conditions on the ideal , we prove an equivalence inequality involving...

A note on the multiplier ideals of monomial ideals

Cheng Gong, Zhongming Tang (2015)

Czechoslovak Mathematical Journal

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Let $𝔞 \subseteq ℂ [x_{1}, ..., x_{n}]$ be a monomial ideal and $𝒥 (𝔞^{c})$ the multiplier ideal of $𝔞$ with coefficient $c$ . Then $𝒥 (𝔞^{c})$ is also a monomial ideal of $ℂ [x_{1}, ..., x_{n}]$ , and the equality $𝒥 (𝔞^{c}) = 𝔞$ implies that $0 < c < n + 1$ . We mainly discuss the problem when $𝒥 (𝔞) = 𝔞$ or $𝒥 (𝔞^{n + 1 - ε}) = 𝔞$ for all $0 < ε < 1$ . It is proved that if $𝒥 (𝔞) = 𝔞$ then $𝔞$ is principal, and if $𝒥 (𝔞^{n + 1 - ε}) = 𝔞$ holds for all $0 < ε < 1$ then $𝔞 = (x_{1}, ..., x_{n})$ . One global result is also obtained. Let $\tilde{𝔞}$ be the ideal sheaf on $ℙ^{n - 1}$ associated with $𝔞$ . Then it is proved that the equality $𝒥 (\tilde{𝔞}) = \tilde{𝔞}$ implies that $\tilde{𝔞}$ is principal.

A co-ideal based identity-summand graph of a commutative semiring

S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let $Γ_{I} (R)$ be a graph with the set of vertices $S_{I} (R) = {x \in R ∖ I : x + y \in I$ for some $y \in R ∖ I}$ , where two distinct vertices $x$ and $y$ are adjacent if and only if $x + y \in I$ . We look at the diameter and girth of this graph. Also we discuss when $Γ_{I} (R)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.

Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

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Let $\sum_{n = 1}^{\infty} a_{n}$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_{n})$ is non-increasing, then $lim_{n \to \infty} n a_{n} = 0$ . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $lim_{n \to \infty} n a_{n} = 0$ ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $ℐ$ -convergence, that is a convergence according to an ideal $ℐ$ of subsets of $ℕ$ . Again, Olivier’s theorem is a consequence...

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 the regularity and defect sequence of monomial and binomial ideals

Keivan Borna, Abolfazl Mohajer (2019)

Czechoslovak Mathematical Journal

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When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$ , with homogeneous maximal ideal $𝔪$ , it is known that for an ideal $I$ of $S$ , the regularity of powers of $I$ becomes eventually a linear function, i.e., $reg (I^{m}) = d m + e$ for $m ≫ 0$ and some integers $d$ , $e$ . This motivates writing $reg (I^{m}) = d m + e_{m}$ for every $m \geq 0$ . The sequence $e_{m}$ , called the of the ideal $I$ , is the subject of much research and its nature is still widely unexplored. We know that $e_{m}$ is eventually constant. In this article, after proving...