On generalized CS-modules

Qingyi Zeng

Displaying similar documents to “On generalized CS-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.

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 ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ ...

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.

Relative tilting modules with respect to a semidualizing module

Maryam Salimi (2019)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, and let $C$ be a semidualizing $R$ -module. The notion of $C$ -tilting $R$ -modules is introduced as the relative setting of the notion of tilting $R$ -modules with respect to $C$ . Some properties of tilting and $C$ -tilting modules and the relations between them are mentioned. It is shown that every finitely generated $C$ -tilting $R$ -module is $C$ -projective. Finally, we investigate some kernel subcategories related to $C$ -tilting modules.

Dual modules and reflexive modules with respect to a semidualizing module

Lixin Mao (2024)

Czechoslovak Mathematical Journal

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Let $C$ be a semidualizing module over a commutative ring. We first investigate the properties of $C$ -dual, $C$ -torsionless and $C$ -reflexive modules. Then we characterize some rings such as coherent rings, $Π$ -coherent rings and FP-injectivity of $C$ using $C$ -dual, $C$ -torsionless and $C$ -reflexive properties of some special modules.

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 $C \otimes_{R} M$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum. ...

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.

A note on generalizations of semisimple modules

Engin Kaynar, Burcu N. Türkmen, Ergül Türkmen (2019)

Commentationes Mathematicae Universitatis Carolinae

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A left module $M$ over an arbitrary ring is called an $ℛ𝒟$ -module (or an $ℛ𝒮$ -module) if every submodule $N$ of $M$ with $Rad (M) \subseteq N$ is a direct summand of (a supplement in, respectively) $M$ . In this paper, we investigate the various properties of $ℛ𝒟$ -modules and $ℛ𝒮$ -modules. We prove that $M$ is an $ℛ𝒟$ -module if and only if $M = Rad (M) \oplus X$ , where $X$ is semisimple. We show that a finitely generated $ℛ𝒮$ -module is semisimple. This gives us the characterization of semisimple rings in terms of $ℛ𝒮$ -modules. We completely determine the structure...

Special modules for $R (PSL (2, q))$

Liufeng Cao, Huixiang Chen (2023)

Czechoslovak Mathematical Journal

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Let $R$ be a fusion ring and $R_{ℂ} : = R \otimes_{ℤ} ℂ$ be the corresponding fusion algebra. We first show that the algebra $R_{ℂ}$ has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, $R_{ℂ}$ admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra $R (PSL (2, q)) : = r (PSL (2, q)) \otimes_{ℤ} ℂ$ up to isomorphism, where $r (PSL (2, q))$ is the...

Generalized tilting modules over ring extension

Zhen Zhang (2019)

Czechoslovak Mathematical Journal

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Let $Γ$ be a ring extension of $R$ . We show the left $Γ$ -module $U = Γ \otimes_{R} C$ with the endmorphism ring End $_{Γ} U = Δ$ is a generalized tilting module when $_{R} C$ is a generalized tilting module under some conditions.

Stratified modules over an extension algebra

Erzsébet Lukács, András Magyar (2018)

Czechoslovak Mathematical Journal

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Let $A$ be a standard Koszul standardly stratified algebra and $X$ an $A$ -module. The paper investigates conditions which imply that the module ${Ext}_{A}^{*} (X)$ over the Yoneda extension algebra $A^{*}$ is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of $A$ is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

$α$ -modules and generalized submodules

Rafiquddin Rafiquddin, Ayazul Hasan, Mohammad Fareed Ahmad (2019)

Communications in Mathematics

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A QTAG-module $M$ is an $α$ -module, where $α$ is a limit ordinal, if $M / H_{β} (M)$ is totally projective for every ordinal $β < α$ . In the present paper $α$ -modules are studied with the help of $α$ -pure submodules, $α$ -basic submodules, and $α$ -large submodules. It is found that an $α$ -closed $α$ -module is an $α$ -injective. For any ordinal $ω \leq α \leq ω_{1}$ we prove that an $α$ -large submodule $L$ of an $ω_{1}$ -module $M$ is summable if and only if $M$ is summable.

Ding projective and Ding injective modules over trivial ring extensions

Lixin Mao (2023)

Czechoslovak Mathematical Journal

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Let $R ⋉ M$ be a trivial extension of a ring $R$ by an $R$ - $R$ -bimodule $M$ such that $M_{R}$ , $_{R} M$ , ${(R, 0)}_{R ⋉ M}$ and $_{R ⋉ M} (R, 0)$ have finite flat dimensions. We prove that $(X, α)$ is a Ding projective left $R ⋉ M$ -module if and only if the sequence $M \otimes_{R} M \otimes_{R} X \overset{M \otimes α}{⟶} M \otimes_{R} X \overset{α}{\to} X$ is exact and $coker (α)$ is a Ding projective left $R$ -module. Analogously, we explicitly describe Ding injective $R ⋉ M$ -modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.

Coherence relative to a weak torsion class

Zhanmin Zhu (2018)

Czechoslovak Mathematical Journal

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Let $R$ be a ring. A subclass $𝒯$ of left $R$ -modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let $𝒯$ be a weak torsion class of left $R$ -modules and $n$ a positive integer. Then a left $R$ -module $M$ is called $𝒯$ -finitely generated if there exists a finitely generated submodule $N$ such that $M / N \in 𝒯$ ; a left $R$ -module $A$ is called $(𝒯, n)$ -presented if there exists an exact sequence of left $R$ -modules $0 ⟶ K_{n - 1} ⟶ F_{n - 1} ⟶ \dots ⟶ F_{1} ⟶ F_{0} ⟶ M ⟶ 0$ such that $F_{0}, \dots, F_{n - 1}$ are finitely generated free and $K_{n - 1}$ is $𝒯$ -finitely generated;...