Displaying similar documents to “FC-modules with an application to cotorsion pairs”

( n , d ) -injective covers, n -coherent rings, and ( n , d ) -rings

Weiqing Li, Baiyu Ouyang (2014)

Czechoslovak Mathematical Journal

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It is known that a ring R is left Noetherian if and only if every left R -module has an injective (pre)cover. We show that ( 1 ) if R is a right n -coherent ring, then every right R -module has an ( n , d ) -injective (pre)cover; ( 2 ) if R is a ring such that every ( n , 0 ) -injective right R -module is n -pure extending, and if every right R -module has an ( n , 0 ) -injective cover, then R is right n -coherent. As applications of these results, we give some characterizations of ( n , d ) -rings, von Neumann regular rings and semisimple...

⊕ -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...

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 ∈ Spec ( R ) with depth ( R 𝔭 ) ≤ 1 , and G- dim R 𝔭 ( M 𝔭 ) ≤ depth ( R 𝔭 ) - 2 for 𝔭 ∈ Spec ( R ) with depth ( R 𝔭 ) ≥ 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 ≥ 2 we give a characterization of n -Gorenstein rings via Gorenstein dimension of the dual of modules. Finally...

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...