Displaying similar documents to “Pretty cleanness and filter-regular sequences”

Decomposition of finitely generated modules using Fitting ideals

Somayeh Hadjirezaei, Sina Hedayat (2020)

Czechoslovak Mathematical Journal

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Let R be a commutative Noetherian ring and M be a finitely generated R -module. The main result of this paper is to characterize modules whose first nonzero Fitting ideal is a product of maximal ideals of R , in some cases.

On the uniform behaviour of the Frobenius closures of ideals

K. Khashyarmanesh (2007)

Colloquium Mathematicae

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Let be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q() be the smallest positive integer m such that ( F ) [ p m ] = [ p m ] , where F is the Frobenius closure of . This paper is concerned with the question whether the set Q ( [ p m ] ) : m is bounded. We give an affirmative answer in the case that the ideal is generated by an u.s.d-sequence c₁,..., cₙ for R such that (i) the map R / j = 1 n R c j R / j = 1 n R c ² j induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all a s s ( c j , . . . , c j ) , c₁/1,..., cₙ/1 is a R -filter...

Local-global principle for annihilation of general local cohomology

J. Asadollahi, K. Khashyarmanesh, Sh. Salarian (2001)

Colloquium Mathematicae

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Let A be a Noetherian ring, let M be a finitely generated A-module and let Φ be a system of ideals of A. We prove that, for any ideal in Φ, if, for every prime ideal of A, there exists an integer k(), depending on , such that k ( ) kills the general local cohomology module H Φ j ( M ) for every integer j less than a fixed integer n, where Φ : = : Φ , then there exists an integer k such that k H Φ j ( M ) = 0 for every j < n.

Some results on top local cohomology modules with respect to a pair of ideals

Saeed Jahandoust, Reza Naghipour (2020)

Mathematica Bohemica

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Let I and J be ideals of a Noetherian local ring ( R , 𝔪 ) and let M be a nonzero finitely generated R -module. We study the relation between the vanishing of H I , J dim M ( M ) and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian R -module M / J M is equal to its integral closure relative to the Artinian R -module H I , J dim M ( M ) .

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...

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...

Cohomological dimension filtration and annihilators of top local cohomology modules

Ali Atazadeh, Monireh Sedghi, Reza Naghipour (2015)

Colloquium Mathematicae

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Let denote an ideal in a Noetherian ring R, and M a finitely generated R-module. We introduce the concept of the cohomological dimension filtration = M i i = 0 c , where c = cd(,M) and M i denotes the largest submodule of M such that c d ( , M i ) i . Some properties of this filtration are investigated. In particular, if (R,) is local and c = dim M, we are able to determine the annihilator of the top local cohomology module H c ( M ) , namely A n n R ( H c ( M ) ) = A n n R ( M / M c - 1 ) . As a consequence, there exists an ideal of R such that A n n R ( H c ( M ) ) = A n n R ( M / H ( M ) ) . This generalizes the...

Left quotients of a C*-algebra, III: Operators on left quotients

Lawrence G. Brown, Ngai-Ching Wong (2013)

Studia Mathematica

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Let L be a norm closed left ideal of a C*-algebra A. Then the left quotient A/L is a left A-module. In this paper, we shall implement Tomita’s idea about representing elements of A as left multiplications: π p ( a ) ( b + L ) = a b + L . A complete characterization of bounded endomorphisms of the A-module A/L is given. The double commutant π p ( A ) ' ' of π p ( A ) in B(A/L) is described. Density theorems of von Neumann and Kaplansky type are obtained. Finally, a comprehensive study of relative multipliers of A is carried out. ...

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 .

Artinian cofinite modules over complete Noetherian local rings

Behrouz Sadeghi, Kamal Bahmanpour, Jafar A&#039;zami (2013)

Czechoslovak Mathematical Journal

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Let ( R , 𝔪 ) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R -module. In this paper it is shown that if 𝔭 is a prime ideal of R such that dim R / 𝔭 = 1 and ( 0 : M 𝔭 ) is not finitely generated and for each i 2 the R -module Ext R i ( M , R / 𝔭 ) is of finite length, then the R -module Ext R 1 ( M , R / 𝔭 ) is not of finite length. Using this result, it is shown that for all finitely generated R -modules N with Supp ( N ) V ( I ) and for all integers i 0 , the R -modules Ext R i ( N , M ) are of finite length, if and only if, for all finitely generated R -modules...

Characterizations of incidence modules

Naseer Ullah, Hailou Yao, Qianqian Yuan, Muhammad Azam (2024)

Czechoslovak Mathematical Journal

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Let R be an associative ring and M be a left R -module. We introduce the concept of the incidence module I ( X , M ) of a locally finite partially ordered set X over M . We study the properties of I ( X , M ) and give the necessary and sufficient conditions for the incidence module to be an IN-module, -module, nil injective module and nonsingular module, respectively. Furthermore, we show that the class of -modules is closed under direct product and upper triangular matrix modules.

On commutative rings whose maximal ideals are idempotent

Farid Kourki, Rachid Tribak (2019)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a commutative ring R , every noetherian (artinian) R -module is quasi-injective if and only if every noetherian (artinian) R -module is quasi-projective if and only if the class of noetherian (artinian) R -modules is socle-fine if and only if the class of noetherian (artinian) R -modules is radical-fine if and only if every maximal ideal of R is idempotent.

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Kosar Abolfath Beigi, Kamran Divaani-Aazar, Massoud Tousi (2022)

Czechoslovak Mathematical Journal

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Let R be a local ring and C a semidualizing module of R . We investigate the behavior of certain classes of generalized Cohen-Macaulay R -modules under the Foxby equivalence between the Auslander and Bass classes with respect to C . In particular, we show that generalized Cohen-Macaulay R -modules are invariant under this equivalence and if M is a finitely generated R -module in the Auslander class with respect to C such that C R M is surjective Buchsbaum, then M is also surjective Buchsbaum. ...

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 ) ) 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....