Displaying similar documents to “Uppers to zero in R [ x ] and almost principal ideals”

On wsq-primary ideals

Emel Aslankarayiğit Uğurlu, El Mehdi Bouba, Ünsal Tekir, Suat Koç (2023)

Czechoslovak Mathematical Journal

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We introduce weakly strongly quasi-primary (briefly, wsq-primary) ideals in commutative rings. Let R be a commutative ring with a nonzero identity and Q a proper ideal of R . The proper ideal Q is said to be a weakly strongly quasi-primary ideal if whenever 0 ≠ a b ∈ Q for some a , b ∈ R , then a 2 ∈ Q or b ∈ Q . Many examples and properties of wsq-primary ideals are given. Also, we characterize nonlocal Noetherian von Neumann regular rings, fields, nonlocal rings over which every proper ideal is wsq-primary, and zero...

Characterization of irreducible polynomials over a special principal ideal ring

Brahim Boudine (2023)

Mathematica Bohemica

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A commutative ring R with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length e is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length 2 . Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length e .

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 ( a ℓ ) / K ) for every choice of a . We then show that becomes principal in L if and only if every reciprocal prime is not...

Star operations in extensions of integral domains

David F. Anderson, Said El Baghdadi, Muhammad Zafrullah (2010)

Actes des rencontres du CIRM

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An extension D ⊆ R of integral domains is t - (resp., t -) if ( I R ) - 1 = ( I - 1 R ) v (resp., ( I R ) v = ( I v R ) v ) for every nonzero finitely generated fractional ideal I of D . We show that strongly t -compatible implies t -compatible and give examples to show that the converse does not hold. We also indicate situations where strong t -compatibility and its variants show up naturally. In addition, we study integral domains D such that D ⊆ R is strongly t -compatible (resp., t -compatible) for every overring R of D . ...

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...

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 ∈ R such that a b c ∈ I , it is either a b ∈ I or c ∈ 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 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 ⊕ 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...

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

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

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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.

0 -ideals in 0 -distributive posets

Khalid A. Mokbel (2016)

Mathematica Bohemica

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The concept of a 0 -ideal in 0 -distributive posets is introduced. Several properties of 0 -ideals in 0 -distributive posets are established. Further, the interrelationships between 0 -ideals and α -ideals in 0 -distributive posets are investigated. Moreover, a characterization of prime ideals to be 0 -ideals in 0 -distributive posets is obtained in terms of non-dense ideals. It is shown that every 0 -ideal of a 0 -distributive meet semilattice is semiprime. Several counterexamples are discussed. ...