Spectral gabriel topologies and relative singular functors

Pardo, José Luis Gómez

doi:10.1080/00927878508823147

Cited by 21 publications

(10 citation statements)

References 13 publications

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“…In [9], Pardo defined τ -large submodules as a torsion-theoretic analogue of large (essential) submodules. A submodule N of an R-module M is called…”

Section: τ -Closed and τ -Complement Submodulesmentioning

confidence: 99%

“…We write N τ M to denote that N is a τ -large submodule of M . In [9,Remarks p. 35], Pardo showed that in general, the set of τ -large submodules and large submodules of a module are different.…”

Section: τ -Closed and τ -Complement Submodulesmentioning

confidence: 99%

“…We begin with some results about the properties of τ -large submodules from Propositions 2.2, 2.3 and 2.4. in [9]. …”

Section: τ -Closed and τ -Complement Submodulesmentioning

confidence: 99%

“…In the second part of this note, we study the properties of τ -closed submodules defined by Pardo ( [9]) and also give some examples which show that closed and τ -closed submodules are different concepts. We also study the relations between τ -closed and τ -complement submodules.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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On τ-Extending Modules

Çeken

Alkan

2010

Mediterr. J. Math.

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Motivated by [2] and [6], we introduce a generalization of extending (CS) modules by using the concept of τ -large submodule which was defined in [9]. We give some properties of this class of modules and study their relationship with the familiar concepts of τ -closed, τ -complement submodules and the other generalization of extending modules (τ -complemented, τ -CS, s-τ -CS modules). We are also interested in determining when a τ -divisible module is τ -extending. For a τ -extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that M = τ (M ) ⊕ K and K is τ (M )-injective. We also treat when a direct sum of τ -extending modules is τ -extending.Mathematics Subject Classification (2010). Primary 16S90; Secondary 16D50, 16D70.

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“…In [9], Pardo defined τ -large submodules as a torsion-theoretic analogue of large (essential) submodules. A submodule N of an R-module M is called…”

Section: τ -Closed and τ -Complement Submodulesmentioning

confidence: 99%

Section: τ -Closed and τ -Complement Submodulesmentioning

confidence: 99%

“…We begin with some results about the properties of τ -large submodules from Propositions 2.2, 2.3 and 2.4. in [9]. …”

Section: τ -Closed and τ -Complement Submodulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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On τ-Extending Modules

Çeken

Alkan

2010

Mediterr. J. Math.

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Continuous and pf rings of quotients

Hernández

1988

Ring Theory

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Rings of quotients of endomorphism rings

Pardo

1988

Ring Theory

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Introduction.The problem of describing the projective modules R M such that their endomorphism ring S = End(RM) satisfies some specific property has been considered in many papers. A frequent approach consists in defining a module-theoretic analogue of a ring-theoretic property and trying to prove that these modules provide the desired solution. But this approach is somewhat naYve, for in the passage from R to M too many things are lost: in categorical terms R is, among other things, a finitely generated generator of its module category R-mod but M is not. One of the first papers that deals with these questions is [i~]. Therein, Ware succesfully solves the problem for semiperfect and perfect endomorphism rings but the regular modules that he defines give only a sufficient condition (in the finitely generated case) for S to be a (Von Neumann) regular ring, which illustrates the drawbacks of this method. In [7] and [8] a rather general technique has been introduced to find necessary and sufficient conditions on M for S to have a specific property. In fact, since M is not supposed to be a generator, we do not need to work with all the category R-rood, which is replaced by the Grothendieck category o[M ] of all the modules subgenerated by M (i.e., submodules of quotients of direct sums of M). This makes unnecessary the projectivity of M and we will only assume that M is E-quasi-projective (i.e., that all direct sums of copies of M are quasi-projective or, equivalently, M is a projective object of o[M]). If M is, furthermore, finitely generated, then this condition reduces to quasi-projectivity. When M is a finitely generated quasi-projective selfgenerator (M generates all its submodules), then HomR(M,-) induces a category equivalence between ~[M] and S-mod by a result of Fuller [5]. This is a sort of Morita equivalence "relative to M" but the hypotheses on M are still very strong. In general, M is not a generator of a[M] (a self-generator) due to the fact that there are in d[M] modules in which M has zero trace, that is, the class TM= {Xec[M]IHomR(M,X) = O} can have nonzero modules. But we may get a generator in the following way: consider the quotient category ~[M]/T M (in the sense of Gabriel's localization theory [6]) of o[M] modulo the hereditary torsion class T M. Then o[M]/T M is a Grothendieck category, there is an exact canonical functor O:a[M] , c[M]/T M and Q(M) is a projective generator of o[M]/T M whose endomorphism ring as an object of this category is canoni-*Work partially supported by the CAICYT.

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Spectral gabriel topologies and relative singular functors

Cited by 21 publications

References 13 publications

On τ-Extending Modules

On τ-Extending Modules

Continuous and pf rings of quotients

Rings of quotients of endomorphism rings

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