Flatness testing over singular bases

Janusz Adamus; Hadi Seyedinejad

Displaying similar documents to “Flatness testing over singular bases”

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...

Structure of flat covers of injective modules

Sh. Payrovi, M. Akhavizadegan (2003)

Colloquium Mathematicae

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The aim of this paper is to discuss the flat covers of injective modules over a Noetherian ring. Let R be a commutative Noetherian ring and let E be an injective R-module. We prove that the flat cover of E is isomorphic to $\prod_{p \in A t t_{R} (E)} T_{p}$ . As a consequence, we give an answer to Xu’s question [10, 4.4.9]: for a prime ideal p, when does $T_{p}$ appear in the flat cover of E(R/m̲)?

A Generalization of Baer's Lemma

Molly Dunkum (2009)

Czechoslovak Mathematical Journal

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There is a classical result known as Baer’s Lemma that states that an $R$ -module $E$ is injective if it is injective for $R$ . This means that if a map from a submodule of $R$ , that is, from a left ideal $L$ of $R$ to $E$ can always be extended to $R$ , then a map to $E$ from a submodule $A$ of any $R$ -module $B$ can be extended to $B$ ; in other words, $E$ is injective. In this paper, we generalize this result to the category $q_{ω}$ consisting of the representations of an infinite line quiver. This generalization of Baer’s...

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.

Derived dimension via $τ$ -tilting theory

Yingying Zhang (2021)

Czechoslovak Mathematical Journal

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Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support $τ$ -tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given $τ$ -tilting module.

Some results on $G_{C}$ -flat dimension of modules

Ramalingam Udhayakumar, Intan Muchtadi-Alamsyah, Chelliah Selvaraj (2019)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we study some properties of $G_{C}$ -flat $R$ -modules, where $C$ is a semidualizing module over a commutative ring $R$ and we investigate the relation between the $G_{C}$ -yoke with the $C$ -yoke of a module as well as the relation between the $G_{C}$ -flat resolution and the flat resolution of a module over $G F$ -closed rings. We also obtain a criterion for computing the $G_{C}$ -flat dimension of modules.

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.

Some module cohomological properties of Banach algebras

Elham Ilka, Amin Mahmoodi, Abasalt Bodaghi (2020)

Mathematica Bohemica

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We find some relations between module biprojectivity and module biflatness of Banach algebras $𝒜$ and $ℬ$ and their projective tensor product $𝒜 \hat{\otimes} ℬ$ . For some semigroups $S$ , we study module biprojectivity and module biflatness of semigroup algebras $l^{1} (S)$ .

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 ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ ...

$τ$ -supplemented modules and $τ$ -weakly supplemented modules

Muhammet Tamer Koşan (2007)

Archivum Mathematicum

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Given a hereditary torsion theory $τ = (𝕋, 𝔽)$ in Mod- $R$ , a module $M$ is called $τ$ -supplemented if every submodule $A$ of $M$ contains a direct summand $C$ of $M$ with $A / C$ $τ -$ torsion. A submodule $V$ of $M$ is called $τ$ -supplement of $U$ in $M$ if $U + V = M$ and $U \cap V \leq τ (V)$ and $M$ is $τ$ -weakly supplemented if every submodule of $M$ has a $τ$ -supplement in $M$ . Let $M$ be a $τ$ -weakly supplemented module. Then $M$ has a decomposition $M = M_{1} \oplus M_{2}$ where $M_{1}$ is a semisimple module and $M_{2}$ is a module with $τ (M_{2}) \leq_{e} M_{2}$ . Also, it is shown that; any finite sum of $τ$ -weakly...