Pretty cleanness and filter-regular sequences

Somayeh Bandari; Kamran Divaani-Aazar; Ali Soleyman Jahan

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

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 $Q (^{[p^{m}]}) : m \in ℕ ₀$ 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 $R / \sum_{j = 1}^{n} R c_{j} \to R / \sum_{j = 1}^{n} R c ²_{j}$ induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all $\in a s s (c ₁^{j}, . . ., c ₙ^{j})$ , 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 $Φ_{} : =_{} : \in Φ$ , then there exists an integer k such that $^{k} H_{Φ}^{j} (M) = 0$ for every j < n.

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 $c d (, M_{i}) \leq i$ . 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 $A n n_{R} (H_{}^{c} (M)) = A n n_{R} (M / H ⁰_{} (M))$ . 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. ...

Artinian cofinite modules over complete Noetherian local rings

Behrouz Sadeghi, Kamal Bahmanpour, Jafar A'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 $i \geq 2$ 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 $Supp (N) \subseteq V (I)$ and for all integers $i \geq 0$ , the $R$ -modules ${Ext}_{R}^{i} (N, M)$ are of finite length, if and only if, for all finitely generated $R$ -modules...

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....

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.

On generalized CS-modules

Qingyi Zeng (2015)

Czechoslovak Mathematical Journal

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An $𝒮$ -closed submodule of a module $M$ is a submodule $N$ for which $M / N$ is nonsingular. A module $M$ is called a generalized CS-module (or briefly, GCS-module) if any $𝒮$ -closed submodule $N$ of $M$ is a direct summand of $M$ . Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right $R$ -modules are projective if and only if all right $R$ -modules are GCS-modules.

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...

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.

Relative Gorenstein injective covers with respect to a semidualizing module

Elham Tavasoli, Maryam Salimi (2017)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$ -module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_{C}$ -injective module $G$ , the character module $G^{+}$ is $G_{C}$ -flat, then the class ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ ...

Uppers to zero in $R [x]$ and almost principal ideals

Keivan Borna, Abolfazl Mohajer-Naser (2013)

Czechoslovak Mathematical Journal

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Let $R$ be an integral domain with quotient field $K$ and $f (x)$ a polynomial of positive degree in $K [x]$ . In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form $I = f (x) K [x] \cap R [x]$ are almost principal in the following two cases: – $J$ , the ideal generated by the leading coefficients of $I$ , satisfies $J^{- 1} = R$ . – $I^{- 1}$ as the $R [x]$ -submodule of $K (x)$ is of finite type. Furthermore we prove that for $I = f (x) K [x] \cap R [x]$ we have: – $I^{- 1} \cap K [x] = (I :_{K (x)} I)$ . – If there exists...

Testing flatness and computing rank of a module using syzygies

Oswaldo Lezama (2009)

Colloquium Mathematicae

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Using syzygies computed via Gröbner bases techniques, we present algorithms for testing some homological properties for submodules of the free module $A^{m}$ , where A = R[x₁,...,xₙ] and R is a Noetherian commutative ring. We will test if a given submodule M of $A^{m}$ is flat. We will also check if M is locally free of constant dimension. Moreover, we present an algorithm that computes the rank of a flat submodule M of $A^{m}$ and also an algorithm that computes the projective dimension of an arbitrary...