Algebraic bounds on analytic multiplier ideals

Brian Lehmann

Displaying similar documents to “Algebraic bounds on analytic multiplier ideals”

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...

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 $_{κ} λ$ .

$(δ, 2)$ -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...

$α$ -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...

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...

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...

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 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.

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...

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...

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...

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...