Displaying similar documents to “Stratified modules over an extension algebra”

Recollements induced by good (co)silting dg-modules

Rongmin Zhu, Jiaqun Wei (2023)

Czechoslovak Mathematical Journal

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Let U be a dg- A -module, B the endomorphism dg-algebra of U . We know that if U is a good silting object, then there exist a dg-algebra C and a recollement among the derived categories 𝐃 ( C , d ) of C , 𝐃 ( B , d ) of B and 𝐃 ( A , d ) of A . We investigate the condition under which the induced dg-algebra C is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained....

Some results on ( n , d ) -injective modules, ( n , d ) -flat modules and n -coherent rings

Zhanmin Zhu (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let n , d be two non-negative integers. A left R -module M is called ( n , d ) -injective, if Ext d + 1 ( N , M ) = 0 for every n -presented left R -module N . A right R -module V is called ( n , d ) -flat, if Tor d + 1 ( V , N ) = 0 for every n -presented left R -module N . A left R -module M is called weakly n - F P -injective, if Ext n ( N , M ) = 0 for every ( n + 1 ) -presented left R -module N . A right R -module V is called weakly n -flat, if Tor n ( V , N ) = 0 for every ( n + 1 ) -presented left R -module N . In this paper, we give some characterizations and properties of ( n , d ) -injective modules and ( n , d ) -flat modules in...

Relative tilting modules with respect to a semidualizing module

Maryam Salimi (2019)

Czechoslovak Mathematical Journal

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Let R be a commutative Noetherian ring, and let C be a semidualizing R -module. The notion of C -tilting R -modules is introduced as the relative setting of the notion of tilting R -modules with respect to C . Some properties of tilting and C -tilting modules and the relations between them are mentioned. It is shown that every finitely generated C -tilting R -module is C -projective. Finally, we investigate some kernel subcategories related to C -tilting modules.

α -modules and generalized submodules

Rafiquddin Rafiquddin, Ayazul Hasan, Mohammad Fareed Ahmad (2019)

Communications in Mathematics

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A QTAG-module M is an α -module, where α is a limit ordinal, if M / H β ( M ) is totally projective for every ordinal β < α . In the present paper α -modules are studied with the help of α -pure submodules, α -basic submodules, and α -large submodules. It is found that an α -closed α -module is an α -injective. For any ordinal ω α ω 1 we prove that an α -large submodule L of an ω 1 -module M is summable if and only if M is summable.

A note on generalizations of semisimple modules

Engin Kaynar, Burcu N. Türkmen, Ergül Türkmen (2019)

Commentationes Mathematicae Universitatis Carolinae

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A left module M over an arbitrary ring is called an ℛ𝒟 -module (or an ℛ𝒮 -module) if every submodule N of M with Rad ( M ) N is a direct summand of (a supplement in, respectively) M . In this paper, we investigate the various properties of ℛ𝒟 -modules and ℛ𝒮 -modules. We prove that M is an ℛ𝒟 -module if and only if M = Rad ( M ) X , where X is semisimple. We show that a finitely generated ℛ𝒮 -module is semisimple. This gives us the characterization of semisimple rings in terms of ℛ𝒮 -modules. We completely determine the structure...

Coherence relative to a weak torsion class

Zhanmin Zhu (2018)

Czechoslovak Mathematical Journal

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Let R be a ring. A subclass 𝒯 of left R -modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let 𝒯 be a weak torsion class of left R -modules and n a positive integer. Then a left R -module M is called 𝒯 -finitely generated if there exists a finitely generated submodule N such that M / N 𝒯 ; a left R -module A is called ( 𝒯 , n ) -presented if there exists an exact sequence of left R -modules 0 K n - 1 F n - 1 F 1 F 0 M 0 such that F 0 , , F n - 1 are finitely generated free and K n - 1 is 𝒯 -finitely generated;...

Some homological properties of amalgamated modules along an ideal

Hanieh Shoar, Maryam Salimi, Abolfazl Tehranian, Hamid Rasouli, Elham Tavasoli (2023)

Czechoslovak Mathematical Journal

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Let R and S be commutative rings with identity, J be an ideal of S , f : R S be a ring homomorphism, M be an R -module, N be an S -module, and let ϕ : M N be an R -homomorphism. The amalgamation of R with S along J with respect to f denoted by R f J was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of ( R f J ) -module called the amalgamation of M and N along J with respect to ϕ , and denoted by M ϕ J N . We study some homological properties of the ( R f J ) -module M ϕ J N . Among...

Cominimaxness of local cohomology modules

Moharram Aghapournahr (2019)

Czechoslovak Mathematical Journal

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Let R be a commutative Noetherian ring, I an ideal of R . Let t 0 be an integer and M an R -module such that Ext R i ( R / I , M ) is minimax for all i t + 1 . We prove that if H I i ( M ) is FD 1 (or weakly Laskerian) for all i < t , then the R -modules H I i ( M ) are I -cominimax for all i < t and Ext R i ( R / I , H I t ( M ) ) is minimax for i = 0 , 1 . Let N be a finitely generated R -module. We prove that Ext R j ( N , H I i ( M ) ) and Tor j R ( N , H I i ( M ) ) are I -cominimax for all i and j whenever M is minimax and H I i ( M ) is FD 1 (or weakly Laskerian) for all i .

On τ -extending modules

Y. Talebi, R. Mohammadi (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we introduce the concept of τ -extending modules by τ -rational submodules and study some properties of such modules. It is shown that the set of all τ -rational left ideals of R R is a Gabriel filter. An R -module M is called τ -extending if every submodule of M is τ -rational in a direct summand of M . It is proved that M is τ -extending if and only if M = R e j M E ( R / τ ( R ) ) N , such that N is a τ -extending submodule of M . An example is given to show that the direct sum of τ -extending modules need not...

Local cohomology, cofiniteness and homological functors of modules

Kamal Bahmanpour (2022)

Czechoslovak Mathematical Journal

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Let I be an ideal of a commutative Noetherian ring R . It is shown that the R -modules H I j ( M ) are I -cofinite for all finitely generated R -modules M and all j 0 if and only if the R -modules Ext R i ( N , H I j ( M ) ) and Tor i R ( N , H I j ( M ) ) are I -cofinite for all finitely generated R -modules M , N and all integers i , j 0 .

Ding projective and Ding injective modules over trivial ring extensions

Lixin Mao (2023)

Czechoslovak Mathematical Journal

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Let R M be a trivial extension of a ring R by an R - R -bimodule M such that M R , R M , ( R , 0 ) R M and R M ( R , 0 ) have finite flat dimensions. We prove that ( X , α ) is a Ding projective left R M -module if and only if the sequence M R M R X M α M R X α X is exact and coker ( α ) is a Ding projective left R -module. Analogously, we explicitly describe Ding injective R M -modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.

Rings whose nonsingular right modules are R -projective

Yusuf Alagöz, Sinem Benli, Engin Büyükaşık (2021)

Commentationes Mathematicae Universitatis Carolinae

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A right R -module M is called R -projective provided that it is projective relative to the right R -module R R . This paper deals with the rings whose all nonsingular right modules are R -projective. For a right nonsingular ring R , we prove that R R is of finite Goldie rank and all nonsingular right R -modules are R -projective if and only if R is right finitely Σ - C S and flat right R -modules are R -projective. Then, R -projectivity of the class of nonsingular injective right modules is also considered....

On the minimaxness and coatomicness of local cohomology modules

Marzieh Hatamkhani, Hajar Roshan-Shekalgourabi (2022)

Czechoslovak Mathematical Journal

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Let R be a commutative Noetherian ring, I an ideal of R and M an R -module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and 𝒞 -minimaxness of local cohomology modules. We show that if M is a minimax R -module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if n is a nonnegative integer such that ( H I i ( M ) ) 𝔪 is a minimax R 𝔪 -module for all 𝔪 Max ( R ) and for all i < n , then the set Ass R ( H I n ( M ) ) is finite. Also, if H I i ( M ) is...

A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot (2012)

Bulletin de la Société Mathématique de France

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If X is a smooth scheme over a perfect field of characteristic p , and if 𝒟 X ( ) is the sheaf of differential operators on X [7], it is well known that giving an action of 𝒟 X ( ) on an 𝒪 X -module is equivalent to giving an infinite sequence of 𝒪 X -modules descending via the iterates of the Frobenius endomorphism of X [5]. We show that this result can be generalized to any infinitesimal deformation f : X S of a smooth morphism in characteristic p , endowed with Frobenius liftings. We also show that it...

S -depth on Z D -modules and local cohomology

Morteza Lotfi Parsa (2021)

Czechoslovak Mathematical Journal

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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...