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Indecomposable primarily comultiplication modules over a pullback of two Dedekind domains

S. Ebrahimi Atani, F. Esmaeili Khalil Saraei (2010)

Colloquium Mathematicae

We describe all those indecomposable primarily comultiplication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given by R. Ebrahimi Atani and S. Ebrahimi Atani [Algebra Discrete Math. 2009] to a more general primarily comultiplication modules case.

Melkersson condition on Serre subcategories

Reza Sazeedeh, Rasul Rasuli (2016)

Colloquium Mathematicae

Let R be a commutative noetherian ring, let be an ideal of R, and let be a subcategory of the category of R-modules. The condition C , defined for R-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to belong to . In this paper, we define and study the class consisting of all modules satisfying C . If and are ideals of R, we get a necessary and sufficient condition for to satisfy C and C simultaneously. We also find some sufficient...

S -depth on Z D -modules and local cohomology

Morteza Lotfi Parsa (2021)

Czechoslovak Mathematical Journal

Let R be a Noetherian ring, and I and J be two ideals of R . Let S be a Serre subcategory of the category of R -modules satisfying the condition C I and M be a Z D -module. As a generalization of the S - depth ( I , M ) and depth ( I , J , M ) , the S - depth of ( I , J ) on M is defined as S - depth ( I , J , M ) = inf { S - depth ( 𝔞 , M ) : 𝔞 W ˜ ( I , J ) } , and some properties of this concept are investigated. The relations between S - depth ( I , J , M ) and H I , J i ( M ) are studied, and it is proved that S - depth ( I , J , M ) = inf { i : H I , J i ( M ) S } , where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with...

The torsion theory and the Melkersson condition

Takeshi Yoshizawa (2020)

Czechoslovak Mathematical Journal

We consider a generalization of the notion of torsion theory, which is associated with a Serre subcategory over a commutative Noetherian ring. In 2008 Aghapournahr and Melkersson investigated the question of when local cohomology modules belong to a Serre subcategory of the module category. In their study, the notion of Melkersson condition was defined as a suitable condition in local cohomology theory. One of our purposes in this paper is to show how naturally the concept of Melkersson condition...

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