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...

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...

Matrix rings with summand intersection property

F. Karabacak, Adnan Tercan (2003)

Czechoslovak Mathematical Journal

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A ring $R$ has right SIP (SSP) if the intersection (sum) of two direct summands of $R$ is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of $R$ by $M$ has SIP if and only if $R$ has SIP and $(1 - e) M e = 0$ for every idempotent $e$ in $R$ . Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.