Displaying similar documents to “A family of noetherian rings with their finite length modules under control”

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.

⊕ -cofinitely supplemented modules

H. Çalışıcı, A. Pancar (2004)

Czechoslovak Mathematical Journal

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Let R be a ring and M a right R -module. M is called ⊕ -cofinitely supplemented if every submodule N of M with M N finitely generated has a supplement that is a direct summand of M . In this paper various properties of the ⊕ -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of ⊕ -cofinitely supplemented modules is ⊕ -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R -module is ⊕ -cofinitely supplemented. In addition, if M has the...

Wakamatsu tilting modules with finite injective dimension

Guoqiang Zhao, Lirong Yin (2013)

Czechoslovak Mathematical Journal

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Let R be a left Noetherian ring, S a right Noetherian ring and R ω a Wakamatsu tilting module with S = End ( R ω ) . We introduce the notion of the ω -torsionfree dimension of finitely generated R -modules and give some criteria for computing it. For any n ≥ 0 , we prove that l . id R ( ω ) = r . id S ( ω ) ≤ n if and only if every finitely generated left R -module and every finitely generated right S -module have ω -torsionfree dimension at most n , if and only if every finitely generated left R -module (or right S -module) has generalized Gorenstein...

Artinian cofinite modules over complete Noetherian local rings

Behrouz Sadeghi, Kamal Bahmanpour, Jafar A'zami (2013)

Czechoslovak Mathematical Journal

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Let ( R , 𝔪 ) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R -module. In this paper it is shown that if 𝔭 is a prime ideal of R such that dim R / 𝔭 = 1 and ( 0 : M 𝔭 ) is not finitely generated and for each i ≥ 2 the R -module Ext R i ( M , R / 𝔭 ) is of finite length, then the R -module Ext R 1 ( M , R / 𝔭 ) is not of finite length. Using this result, it is shown that for all finitely generated R -modules N with Supp ( N ) ⊆ V ( I ) and for all integers i ≥ 0 , the R -modules Ext R i ( N , M ) are of finite length, if and only if, for all finitely generated R -modules...