A characterization of the meager ideal

Piotr Zakrzewski

Displaying similar documents to “A characterization of the meager ideal”

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.

The ideal (a) is not $G_{δ}$ generated

Marta Frankowska, Andrzej Nowik (2011)

Colloquium Mathematicae

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We prove that the ideal (a) defined by the density topology is not $G_{δ}$ generated. This answers a question of Z. Grande and E. Strońska.

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....

On the Noether exponent

Anna Stasica (2003)

Annales Polonici Mathematici

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We obtain, in a simple way, an estimate for the Noether exponent of an ideal I without embedded components (i.e. we estimate the smallest number μ such that ${(r a d I)}^{μ} \subset I$ ).

On the ideal class groups of the maximal real subfields of number fields with all roots of unity

Masato Kurihara (1999)

Journal of the European Mathematical Society

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In this paper, for a totally real number field $k$ we show the ideal class group of $k {(\cup_{n > 0} μ_{n})}^{+}$ is trivial. We also study the $p$ -component of the ideal class group of the cyclotomic $𝐙_{p}$ -extension.

An ideal characterization of when a subspace of certain Banach spaces has the metric compact approximation property

J. Cabello, E. Nieto (1998)

Studia Mathematica

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C.-M. Cho and W. B. Johnson showed that if a subspace E of $ℓ_{p}$ , 1 < p < ∞, has the compact approximation property, then K(E) is an M-ideal in ℒ(E). We prove that for every r,s ∈ ]0,1] with $r^{2} + s^{2} < 1$ , the James space can be provided with an equivalent norm such that an arbitrary subspace E has the metric compact approximation property iff there is a norm one projection P on ℒ(E)* with Ker P = K(E)⊥ satisfying ∥⨍∥ ≥ r∥Pf∥ + s∥φ - Pf∥ ∀⨍ ∈ ℒ(E)*. A similar result is proved for subspaces of...

Calculation of the avoiding ideal for $Σ^{1, 1}$

Tamás Terpai (2009)

Banach Center Publications

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We calculate the mapping $H * (B O; ℤ ₂) \to H * (K^{1, 0}; ℤ ₂)$ and obtain a generating system of its kernel. As a corollary, bounds on the codimension of fold maps from real projective spaces to Euclidean space are calculated and the rank of a singular bordism group is determined.

More on cardinal invariants of analytic $P$ -ideals

Barnabás Farkas, Lajos Soukup (2009)

Commentationes Mathematicae Universitatis Carolinae

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Given an ideal $ℐ$ on $ω$ let $𝔞 (ℐ)$ ( $\bar{𝔞} (ℐ)$ ) be minimum of the cardinalities of infinite (uncountable) maximal $ℐ$ -almost disjoint subsets of ${[ω]}^{ω}$ . We show that $𝔞 (ℐ_{h}) > ω$ if $ℐ_{h}$ is a summable ideal; but $𝔞 (𝒵_{\vec{μ}}) = ω$ for any tall density ideal $𝒵_{\vec{μ}}$ including the density zero ideal $𝒵$ . On the other hand, you have $𝔟 \leq \bar{𝔞} (ℐ)$ for any analytic $P$ -ideal $ℐ$ , and $\bar{𝔞} (𝒵_{\vec{μ}}) \leq 𝔞$ for each density ideal $𝒵_{\vec{μ}}$ . For each ideal $ℐ$ on $ω$ denote $𝔟_{ℐ}$ and $𝔡_{ℐ}$ the unbounding and dominating numbers of $〈 ω^{ω}, \leq_{ℐ} 〉$ where $f \leq_{ℐ} g$ iff ${n \in ω : f (n) > g (n)} \in ℐ$ . We show that $𝔟_{ℐ} = 𝔟$ and $𝔡_{ℐ} = 𝔡$ for each analytic $P$ -ideal $ℐ$ . Given a Borel...

Wild primes of a self-equivalence of a number field

Alfred Czogała, Beata Rothkegel (2014)

Acta Arithmetica

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Let K be a number field. Assume that the 2-rank of the ideal class group of K is equal to the 2-rank of the narrow ideal class group of K. Moreover, assume K has a unique dyadic prime and the class of is a square in the ideal class group of K. We prove that if ₁,...,ₙ are finite primes of K such that ∙ the class of $_{i}$ is a square in the ideal class group of K for every i ∈ 1,...,n, ∙ -1 is a local square at $_{i}$ for every nondyadic $_{i} \in ₁, . . ., ₙ$ , then ₁,...,ₙ is the wild set of some self-equivalence...

The ideal boundary of a domain in $C^{n}$

Maciej Skwarczyński (1981)

Annales Polonici Mathematici

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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 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.

Eigenspaces of the ideal class group

Cornelius Greither, Radan Kučera (2014)

Annales de l’institut Fourier

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The aim of this paper is to prove an analog of Gras’ conjecture for an abelian field $F$ and an odd prime $p$ dividing the degree $[F : ℚ]$ assuming that the $p$ -part of $Gal (F / ℚ)$ group is cyclic.

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...

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...

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...

Making sense of capitulation: reciprocal primes

David Folk (2016)

Acta Arithmetica

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Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and $a_{}$ any generator of the principal ideal $^{ℓ}$ . We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates $G a l (K (\sqrt[ℓ]{a_{}}) / K)$ for every choice of $a_{}$ . We then show that becomes principal in L if and only if every reciprocal prime is not...