The ideal (a) is not 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 generated. This answers a question of Z. Grande and E. Strońska.
Marta Frankowska, Andrzej Nowik (2011)
Colloquium Mathematicae
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We prove that the ideal (a) defined by the density topology is not generated. This answers a question of Z. Grande and E. Strońska.
F. Azarpanah, O. A. S. Karamzadeh, S. Rahmati (2008)
Colloquium Mathematicae
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Let be the socle of C(X). It is shown that each prime ideal in 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 , 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....
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 ).
Piotr Zakrzewski (2015)
Commentationes Mathematicae Universitatis Carolinae
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We give a classical proof of the theorem stating that the -ideal of meager sets is the unique -ideal on a Polish group, generated by closed sets which is invariant under translations and ergodic.
Tamás Terpai (2009)
Banach Center Publications
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We calculate the mapping 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.
Masato Kurihara (1999)
Journal of the European Mathematical Society
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In this paper, for a totally real number field we show the ideal class group of is trivial. We also study the -component of the ideal class group of the cyclotomic -extension.
S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel (2015)
Commentationes Mathematicae Universitatis Carolinae
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Let be a strong co-ideal of a commutative semiring with identity. Let be a graph with the set of vertices for some , where two distinct vertices and are adjacent if and only if . We look at the diameter and girth of this graph. Also we discuss when is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.
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 (respectively, ) acting on the operators of the surjective (respectively, injective) hull of such that (respectively, ) if and only if the operator T is -compact (respectively, injectively -compact). Under certain conditions on the ideal , we prove an equivalence inequality involving...
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 and an odd prime dividing the degree assuming that the -part of group is cyclic.
Ali Yassine, Mohammad Javad Nikmehr, Reza Nikandish (2024)
Czechoslovak Mathematical Journal
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Let be a commutative ring with identity. We study the concept of strongly 1-absorbing primary ideals which is a generalization of -ideals and a subclass of -absorbing primary ideals. A proper ideal of is called strongly 1-absorbing primary if for all nonunit elements such that , it is either or . Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings over which every semi-primary ideal is strongly 1-absorbing primary, and rings over which...
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 , 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 , 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...
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 is a square in the ideal class group of K for every i ∈ 1,...,n, ∙ -1 is a local square at for every nondyadic , then ₁,...,ₙ is the wild set of some self-equivalence...
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 (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in , including one that has not been studied before. The proofs use various methods...
Khalid A. Mokbel (2015)
Mathematica Bohemica
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The concept of -ideals in posets is introduced. Several properties of -ideals in -distributive posets are studied. Characterization of prime ideals to be -ideals in -distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal of a -distributive poset is non-dense, then is an -ideal. Moreover, it is shown that the set of all -ideals of a poset with forms a complete lattice. A result analogous to separation theorem for...
Mehrdad Nasernejad, Kazem Khashyarmanesh, Leslie G. Roberts, Jonathan Toledo (2022)
Czechoslovak Mathematical Journal
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Let be an ideal in a commutative Noetherian ring . Then the ideal has the strong persistence property if and only if for all , and has the symbolic strong persistence property if and only if for all , where denotes the th symbolic power of . 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...
Khalid A. Mokbel (2016)
Mathematica Bohemica
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The concept of a -ideal in -distributive posets is introduced. Several properties of -ideals in -distributive posets are established. Further, the interrelationships between -ideals and -ideals in -distributive posets are investigated. Moreover, a characterization of prime ideals to be -ideals in -distributive posets is obtained in terms of non-dense ideals. It is shown that every -ideal of a -distributive meet semilattice is semiprime. Several counterexamples are discussed. ...
Dongyang Chen, William B. Johnson, Bentuo Zheng (2014)
Studia Mathematica
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We give a corrected proof of Theorem 2.10 in our paper “Commutators on ” [Studia Math. 206 (2011), 175-190] for the case 1 < q < p < ∞. The case when 1 = q < p < ∞ remains open. As a consequence, the Main Theorem and Corollary 2.17 in that paper are only valid for 1 < p,q < ∞.
Thomas Vils Pedersen (2004)
Studia Mathematica
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For 0 < γ ≤ 1, let 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 be the closed ideal in consisting of those functions for which (i) f = 0 on E, (ii) as d(z,E),d(w,E) → 0, (iii) . Also, for a closed ideal I in , let = z ∈ : f(z) = 0 for every f ∈ I and let be the greatest common divisor of the inner parts of non-zero functions...
Verónica Dimant (2011)
Studia Mathematica
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We study the problem of whether , 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 is an M-ideal in (ⁿE), then coincides with (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if and (E) is an...
Alain Faisant, Georges Grekos, Ladislav Mišík (2016)
Mathematica Bohemica
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Let be a convergent series of positive real numbers. L. Olivier proved that if the sequence is non-increasing, then . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having ; 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 -convergence, that is a convergence according to an ideal of subsets of . Again, Olivier’s theorem is a consequence...