EUDML | Simple modules over Auslander regular rings

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Displaying similar documents to “Simple modules over Auslander regular rings”

On modules and rings with the restricted minimum condition

M. Tamer Koşan, Jan Žemlička (2015)

Colloquium Mathematicae

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A module M satisfies the restricted minimum condition if M/N is artinian for every essential submodule N of M. A ring R is called a right RM-ring whenever $R_{R}$ satisfies the restricted minimum condition as a right module. We give several structural necessary conditions for particular classes of RM-rings. Furthermore, a commutative ring R is proved to be an RM-ring if and only if R/Soc(R) is noetherian and every singular module is semiartinian.

Strong separativity over exchange rings

Huanyin Chen (2008)

Czechoslovak Mathematical Journal

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An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$ -modules $A$ and $B$ , $A \oplus A ≅ A \oplus B \Rightarrow A ≅ B$ . We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$ , $a S + b S = S$ implies that there exist $u, v \in S$ such that $a u = b v$ and $S u + S v = S$ if and only if for any corner $S$ of $R$ , $a S + b S = S$ implies that there exists a right invertible matrix $(\begin{matrix} a & b \\ * \end{matrix}) \in M_{2} (S)$ . The dual assertions are also proved.

A family of noetherian rings with their finite length modules under control

Markus Schmidmeier (2002)

Czechoslovak Mathematical Journal

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We investigate the category $mod Λ$ of finite length modules over the ring $Λ = A \otimes_{k} Σ$ , where $Σ$ is a V-ring, i.e. a ring for which every simple module is injective, $k$ a subfield of its centre and $A$ an elementary $k$ -algebra. Each simple module $E_{j}$ gives rise to a quasiprogenerator $P_{j} = A \otimes E_{j}$ . By a result of K. Fuller, $P_{j}$ induces a category equivalence from which we deduce that $mod Λ ≃ ∐_{j} b a d h b o x P_{j}$ . As a consequence we can (1) construct for each elementary $k$ -algebra $A$ over a finite field $k$ a nonartinian noetherian ring $Λ$ such that $mod A ≃ mod Λ$ ,...

$k$ -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...

Commutative rings whose certain modules decompose into direct sums of cyclic submodules

Farid Kourki, Rachid Tribak (2023)

Czechoslovak Mathematical Journal

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We provide some characterizations of rings $R$ for which every (finitely generated) module belonging to a class $𝒞$ of $R$ -modules is a direct sum of cyclic submodules. We focus on the cases, where the class $𝒞$ is one of the following classes of modules: semiartinian modules, semi-V-modules, V-modules, coperfect modules and locally supplemented modules.

A generalization of the Auslander transpose and the generalized Gorenstein dimension

Yuxian Geng (2013)

Czechoslovak Mathematical Journal

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Let $R$ be a left and right Noetherian ring and $C$ a semidualizing $R$ -bimodule. We introduce a transpose ${Tr}_{c} M$ of an $R$ -module $M$ with respect to $C$ which unifies the Auslander transpose and Huang’s transpose, see Z. Y. Huang, On a generalization of the Auslander-Bridger transpose, Comm. Algebra 27 (1999), 5791–5812, in the two-sided Noetherian setting, and use ${Tr}_{c} M$ to develop further the generalized Gorenstein dimension with respect to $C$ . Especially, we generalize the Auslander-Bridger formula to...