A variant theory for the Gorenstein flat dimension

Samir Bouchiba

Displaying similar documents to “A variant theory for the Gorenstein flat dimension”

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)?

$k$ -torsionless modules with finite Gorenstein dimension

Maryam Salimi, Elham Tavasoli, Siamak Yassemi (2012)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring. It is shown that the finitely generated $R$ -module $M$ with finite Gorenstein dimension is reflexive if and only if $M_{𝔭}$ is reflexive for $𝔭 \in Spec (R)$ with $depth (R_{𝔭}) \leq 1$ , and ${G- dim}_{R_{𝔭}} (M_{𝔭}) \leq depth (R_{𝔭}) - 2$ for $𝔭 \in Spec (R)$ with $depth (R_{𝔭}) \geq 2$ . This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for $n \geq 2$ we give a characterization of $n$ -Gorenstein rings via Gorenstein dimension of the dual of modules. Finally...

On the structure of sequentially Cohen-Macaulay bigraded modules

Leila Parsaei Majd, Ahad Rahimi (2015)

Czechoslovak Mathematical Journal

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Let $K$ be a field and $S = K [x_{1}, ..., x_{m}, y_{1}, ..., y_{n}]$ be the standard bigraded polynomial ring over $K$ . In this paper, we explicitly describe the structure of finitely generated bigraded “sequentially Cohen-Macaulay” $S$ -modules with respect to $Q = (y_{1}, ..., y_{n})$ . Next, we give a characterization of sequentially Cohen-Macaulay modules with respect to $Q$ in terms of local cohomology modules. Cohen-Macaulay modules that are sequentially Cohen-Macaulay with respect to $Q$ are considered.

Wakamatsu tilting modules with finite injective dimension

Guoqiang Zhao, Lirong Yin (2013)

Czechoslovak Mathematical Journal

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Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and $_{R} ω$ a Wakamatsu tilting module with $S = End (_{R} ω)$ . We introduce the notion of the $ω$ -torsionfree dimension of finitely generated $R$ -modules and give some criteria for computing it. For any $n \geq 0$ , we prove that ${l . id}_{R} (ω) = {r . id}_{S} (ω) \leq n$ if and only if every finitely generated left $R$ -module and every finitely generated right $S$ -module have $ω$ -torsionfree dimension at most $n$ , if and only if every finitely generated left $R$ -module (or right $S$ -module) has generalized Gorenstein...

$n$ - $gr$ -coherent rings and Gorenstein graded modules

Mostafa Amini, Driss Bennis, Soumia Mamdouhi (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a graded ring and $n \geq 1$ be an integer. We introduce and study the notions of Gorenstein $n$ -FP-gr-injective and Gorenstein $n$ -gr-flat modules by using the notion of special finitely presented graded modules. On $n$ -gr-coherent rings, we investigate the relationships between Gorenstein $n$ -FP-gr-injective and Gorenstein $n$ -gr-flat modules. Among other results, we prove that any graded module in $R$ -gr (or gr- $R$ ) admits a Gorenstein $n$ -FP-gr-injective (or Gorenstein $n$ -gr-flat) cover and preenvelope,...

On co-Gorenstein modules, minimal flat resolutions and dual Bass numbers

Zahra Heidarian, Hossein Zakeri (2015)

Colloquium Mathematicae

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The dual of a Gorenstein module is called a co-Gorenstein module, defined by Lingguang Li. In this paper, we prove that if R is a local U-ring and M is an Artinian R-module, then M is a co-Gorenstein R-module if and only if the complex $H o m_{R ̂} ((, R ̂), M)$ is a minimal flat resolution for M when we choose a suitable triangular subset on R̂. Moreover we characterize the co-Gorenstein modules over a local U-ring and Cohen-Macaulay local U-ring.

On Cohen-Macaulay rings

Edgar E. Enochs, Jenda M. G. Overtoun (1994)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we use a characterization of $R$ -modules $N$ such that $f d_{R} N = p d_{R} N$ to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting $N$ to be the $d t h$ local cohomology functor of $R$ with respect to the maximal ideal where $d$ is the Krull dimension of $R$ .

On torsion Gorenstein injective modules

Okyeon Yi (1998)

Archivum Mathematicum

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In this paper, we define Gorenstein injective rings, Gorenstein injective modules and their envelopes. The main topic of this paper is to show that if $D$ is a Gorenstein integral domain and $M$ is a left $D$ -module, then the torsion submodule $t G M$ of Gorenstein injective envelope $G M$ of $M$ is also Gorenstein injective. We can also show that if $M$ is a torsion $D$ -module of a Gorenstein injective integral domain $D$ , then the Gorenstein injective envelope $G M$ of $M$ is torsion.