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Displaying similar documents to “Weak n -injective and weak n -fat modules”

Relative weak derived functors

Panneerselvam Prabakaran (2020)

Commentationes Mathematicae Universitatis Carolinae

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Let R be a ring, n a fixed non-negative integer, 𝒲 the class of all left R -modules with weak injective dimension at most n , and 𝒲 the class of all right R -modules with weak flat dimension at most n . Using left (right) 𝒲 -resolutions and the left derived functors of Hom we study the weak injective dimensions of modules and rings. Also we prove that - - is right balanced on R × R by 𝒲 × 𝒲 , and investigate the global right 𝒲 -dimension of R by right derived functors of .

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....

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.

Eventually semisimple weak F I -extending modules

Figen Takıl Mutlu, Adnan Tercan, Ramazan Yaşar (2023)

Mathematica Bohemica

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In this article, we study modules with the weak F I -extending property. We prove that if M satisfies weak F I -extending, pseudo duo, C 3 properties and M / Soc M has finite uniform dimension then M decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the weak F I -extending, pseudo duo, C 3 properties and ascending (or descending) chain condition on essential submodules then M = M 1 M 2 for some semisimple submodule M 1 and Noetherian (or...

Some results on G C -flat dimension of modules

Ramalingam Udhayakumar, Intan Muchtadi-Alamsyah, Chelliah Selvaraj (2019)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we study some properties of G C -flat R -modules, where C is a semidualizing module over a commutative ring R and we investigate the relation between the G C -yoke with the C -yoke of a module as well as the relation between the G C -flat resolution and the flat resolution of a module over G F -closed rings. We also obtain a criterion for computing the G C -flat dimension of modules.

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...

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...

Stratified modules over an extension algebra

Erzsébet Lukács, András Magyar (2018)

Czechoslovak Mathematical Journal

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Let A be a standard Koszul standardly stratified algebra and X an A -module. The paper investigates conditions which imply that the module Ext A * ( X ) over the Yoneda extension algebra A * is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of A is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

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....

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;...

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.

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.