A variant theory for the Gorenstein flat dimension

Samir Bouchiba

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

Gorenstein star modules and Gorenstein tilting modules

Peiyu Zhang (2021)

Czechoslovak Mathematical Journal

Similarity:

We introduce the notion of Gorenstein star modules and obtain some properties and a characterization of them. We mainly give the relationship between $n$ -Gorenstein star modules and $n$ -Gorenstein tilting modules, see L. Yan, W. Li, B. Ouyang (2016), and a new characterization of $n$ -Gorenstein tilting modules.

Structure of flat covers of injective modules

Sh. Payrovi, M. Akhavizadegan (2003)

Colloquium Mathematicae

Similarity:

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)?

$n$ -strongly Gorenstein graded modules

Zenghui Gao, Jie Peng (2019)

Czechoslovak Mathematical Journal

Similarity:

Let $R$ be a graded ring and $n \geq 1$ an integer. We introduce and study $n$ -strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that $n$ -strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be $m$ -strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever $n > m$ . Many properties of the $n$ -strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized....

$k$ -torsionless modules with finite Gorenstein dimension

Maryam Salimi, Elham Tavasoli, Siamak Yassemi (2012)

Czechoslovak Mathematical Journal

Similarity:

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

Similarity:

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.

Weak dimensions and Gorenstein weak dimensions of group rings

Yueming Xiang (2021)

Czechoslovak Mathematical Journal

Similarity:

Let $K$ be a field, and let $G$ be a group. In the present paper, we investigate when the group ring $K [G]$ has finite weak dimension and finite Gorenstein weak dimension. We give some analogous versions of Serre’s theorem for the weak dimension and the Gorenstein weak dimension.

Wakamatsu tilting modules with finite injective dimension

Guoqiang Zhao, Lirong Yin (2013)

Czechoslovak Mathematical Journal

Similarity:

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

Similarity:

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

Similarity:

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.

Gorenstein projective complexes with respect to cotorsion pairs

Renyu Zhao, Pengju Ma (2019)

Czechoslovak Mathematical Journal

Similarity:

Let $(𝒜, ℬ)$ be a complete and hereditary cotorsion pair in the category of left $R$ -modules. In this paper, the so-called Gorenstein projective complexes with respect to the cotorsion pair $(𝒜, ℬ)$ are introduced. We show that these complexes are just the complexes of Gorenstein projective modules with respect to the cotorsion pair $(𝒜, ℬ)$ . As an application, we prove that both the Gorenstein projective modules with respect to cotorsion pairs and the Gorenstein projective complexes with respect to cotorsion...