Dual modules and reflexive modules with respect to a semidualizing module

Lixin Mao

Displaying similar documents to “Dual modules and reflexive modules with respect to a semidualizing 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.

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...

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.

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 ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ ...

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.

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.

Some results on $(n, d)$ -injective modules, $(n, d)$ -flat modules and $n$ -coherent rings

Zhanmin Zhu (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $n, d$ be two non-negative integers. A left $R$ -module $M$ is called $(n, d)$ -injective, if ${Ext}^{d + 1} (N, M) = 0$ for every $n$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called $(n, d)$ -flat, if ${Tor}_{d + 1} (V, N) = 0$ for every $n$ -presented left $R$ -module $N$ . A left $R$ -module $M$ is called weakly $n$ - $F P$ -injective, if ${Ext}^{n} (N, M) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called weakly $n$ -flat, if ${Tor}_{n} (V, N) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . In this paper, we give some characterizations and properties of $(n, d)$ -injective modules and $(n, d)$ -flat modules in...

Recollements induced by good (co)silting dg-modules

Rongmin Zhu, Jiaqun Wei (2023)

Czechoslovak Mathematical Journal

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Let $U$ be a dg- $A$ -module, $B$ the endomorphism dg-algebra of $U$ . We know that if $U$ is a good silting object, then there exist a dg-algebra $C$ and a recollement among the derived categories $𝐃 (C, d)$ of $C$ , $𝐃 (B, d)$ of $B$ and $𝐃 (A, d)$ of $A$ . We investigate the condition under which the induced dg-algebra $C$ is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained....

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;...

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. ...

$α$ -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.

Rings whose nonsingular right modules are $R$ -projective

Yusuf Alagöz, Sinem Benli, Engin Büyükaşık (2021)

Commentationes Mathematicae Universitatis Carolinae

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A right $R$ -module $M$ is called $R$ -projective provided that it is projective relative to the right $R$ -module $R_{R}$ . This paper deals with the rings whose all nonsingular right modules are $R$ -projective. For a right nonsingular ring $R$ , we prove that $R_{R}$ is of finite Goldie rank and all nonsingular right $R$ -modules are $R$ -projective if and only if $R$ is right finitely $Σ$ - $C S$ and flat right $R$ -modules are $R$ -projective. Then, $R$ -projectivity of the class of nonsingular injective right modules is also considered....