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Displaying similar documents to “A spectral mapping theorem for Banach modules”

On generalized CS-modules

Qingyi Zeng (2015)

Czechoslovak Mathematical Journal

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An 𝒮 -closed submodule of a module M is a submodule N for which M / N is nonsingular. A module M is called a generalized CS-module (or briefly, GCS-module) if any 𝒮 -closed submodule N of M is a direct summand of M . Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right R -modules are projective if and only if all right R -modules are GCS-modules.

Relative Gorenstein injective covers with respect to a semidualizing module

Elham Tavasoli, Maryam Salimi (2017)

Czechoslovak Mathematical Journal

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Let R be a commutative Noetherian ring and let C be a semidualizing R -module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to C which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every G C -injective module G , the character module G + is G C -flat, then the class 𝒢ℐ C ( R ) 𝒜 C ( R ) is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class 𝒢ℐ C ( R ) 𝒜 C ( R ) ...

Characterizations of incidence modules

Naseer Ullah, Hailou Yao, Qianqian Yuan, Muhammad Azam (2024)

Czechoslovak Mathematical Journal

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Let R be an associative ring and M be a left R -module. We introduce the concept of the incidence module I ( X , M ) of a locally finite partially ordered set X over M . We study the properties of I ( X , M ) and give the necessary and sufficient conditions for the incidence module to be an IN-module, -module, nil injective module and nonsingular module, respectively. Furthermore, we show that the class of -modules is closed under direct product and upper triangular matrix modules.

Derived dimension via τ -tilting theory

Yingying Zhang (2021)

Czechoslovak Mathematical Journal

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Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support τ -tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given τ -tilting module.

Dual modules and reflexive modules with respect to a semidualizing module

Lixin Mao (2024)

Czechoslovak Mathematical Journal

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Let C be a semidualizing module over a commutative ring. We first investigate the properties of C -dual, C -torsionless and C -reflexive modules. Then we characterize some rings such as coherent rings, Π -coherent rings and FP-injectivity of C using C -dual, C -torsionless and C -reflexive properties of some special modules.

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Kosar Abolfath Beigi, Kamran Divaani-Aazar, Massoud Tousi (2022)

Czechoslovak Mathematical Journal

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Let R be a local ring and C a semidualizing module of R . We investigate the behavior of certain classes of generalized Cohen-Macaulay R -modules under the Foxby equivalence between the Auslander and Bass classes with respect to C . In particular, we show that generalized Cohen-Macaulay R -modules are invariant under this equivalence and if M is a finitely generated R -module in the Auslander class with respect to C such that C R M is surjective Buchsbaum, then M is also surjective Buchsbaum. ...

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.

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.

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

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Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets ν ( X ) and ν ( X ) of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

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...

Special modules for R ( PSL ( 2 , q ) )

Liufeng Cao, Huixiang Chen (2023)

Czechoslovak Mathematical Journal

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Let R be a fusion ring and R : = R be the corresponding fusion algebra. We first show that the algebra R has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, R admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra R ( PSL ( 2 , q ) ) : = r ( PSL ( 2 , q ) ) up to isomorphism, where r ( PSL ( 2 , q ) ) is the...

A theorem of Gel'fand-Mazur type

Hung Le Pham (2009)

Studia Mathematica

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Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection a α : α A there exist α ≠ β ∈ such that a α a β A is isomorphic to i = 1 r ( [ X ] / X d i [ X ] ) E , where d , . . . , d r , and E is either X [ X ] / X d [ X ] for some d₀ ∈ ℕ or a 1-dimensional i = 1 r [ X ] / X d i [ X ] -bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

On n -submodules and G . n -submodules

Somayeh Karimzadeh, Javad Moghaderi (2023)

Czechoslovak Mathematical Journal

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We investigate some properties of n -submodules. More precisely, we find a necessary and sufficient condition for every proper submodule of a module to be an n -submodule. Also, we show that if M is a finitely generated R -module and Ann R ( M ) is a prime ideal of R , then M has n -submodule. Moreover, we define the notion of G . n -submodule, which is a generalization of the notion of n -submodule. We find some characterizations of G . n -submodules and we examine the way the aforementioned notions are related...

Some module cohomological properties of Banach algebras

Elham Ilka, Amin Mahmoodi, Abasalt Bodaghi (2020)

Mathematica Bohemica

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We find some relations between module biprojectivity and module biflatness of Banach algebras 𝒜 and and their projective tensor product 𝒜 ^ . For some semigroups S , we study module biprojectivity and module biflatness of semigroup algebras l 1 ( S ) .