On commutative rings whose maximal ideals are idempotent

Farid Kourki; Rachid Tribak

Displaying similar documents to “On commutative rings whose maximal ideals are idempotent”

Characterizations of incidence modules

Naseer Ullah, Hailou Yao, Qianqian Yuan, Muhammad Azam (2024)

Czechoslovak Mathematical Journal

Similarity:

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.

On generalized CS-modules

Qingyi Zeng (2015)

Czechoslovak Mathematical Journal

Similarity:

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.

CF-modules over commutative rings

Ahmed Najim, Mohammed Elhassani Charkani (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $R$ be a commutative ring with unit. We give some criterions for determining when a direct sum of two CF-modules over $R$ is a CF-module. When $R$ is local, we characterize the CF-modules over $R$ whose tensor product is a CF-module.

Dual modules and reflexive modules with respect to a semidualizing module

Lixin Mao (2024)

Czechoslovak Mathematical Journal

Similarity:

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.

Relative tilting modules with respect to a semidualizing module

Maryam Salimi (2019)

Czechoslovak Mathematical Journal

Similarity:

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.

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

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

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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

Similarity:

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 ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ ...

Some results on quasi-t-dual Baer modules

Rachid Tribak, Yahya Talebi, Mehrab Hosseinpour (2023)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $R$ be a ring and let $M$ be an $R$ -module with $S = {End}_{R} (M)$ . Consider the preradical $\bar{Z}$ for the category of right $R$ -modules Mod- $R$ introduced by Y. Talebi and N. Vanaja in 2002 and defined by $\bar{Z} (M) = ⋂ {U \leq M : M / U$ is small in its injective hull $}$ . The module $M$ is called quasi-t-dual Baer if $\sum_{ϕ \in ℑ} ϕ ({\bar{Z}}^{2} (M))$ is a direct summand of $M$ for every two-sided ideal $ℑ$ of $S$ , where ${\bar{Z}}^{2} (M) = \bar{Z} (\bar{Z} (M))$ . In this paper, we show that $M$ is quasi-t-dual Baer if and only if ${\bar{Z}}^{2} (M)$ is a direct summand of $M$ and ${\bar{Z}}^{2} (M)$ is a quasi-dual Baer module. It is also shown that any direct...

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

Similarity:

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. ...

A note on generalizations of semisimple modules

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

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...

Derived dimension via $τ$ -tilting theory

Yingying Zhang (2021)

Czechoslovak Mathematical Journal

Similarity:

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.

Generalized tilting modules over ring extension

Zhen Zhang (2019)

Czechoslovak Mathematical Journal

Similarity:

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.