Cotorsion theories

Bronn, Stephen

doi:10.2140/pjm.1973.48.355

Cited by 2 publications

(2 citation statements)

References 4 publications

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“…Colocalizations have been investigated under various approaches by several authors, for example see [3,5,12].…”

Section: Derivations and Modules Of Coquotientsmentioning

confidence: 99%

Differential torsion theory

Bland

2006

Journal of Pure and Applied Algebra

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Differential torsion theories are introduced and it is shown that for a hereditary torsion theory every derivation on an R-module M has a unique extension to its module of quotients if and only if is a differential torsion theory. Dually, we show that when is cohereditary, every derivation on M can be lifed uniquely to its module of coquotients.The purpose of this paper is to introduce the concept of a differential torsion theory on Mod R and to use this notion to study derivations on modules and their extension to modules of quotients. After obtaining the main result concerning such extensions we turn our attention to the problem lifting derivations on modules to modules of coquotients.Throughout R will denote an associative ring with identity, all modules will be unitary right R-modules and Mod R will denote the category of unitary right R-modules. A functionIf is a derivation on R and M is an R-module, then a function d :for all x, y ∈ M and all a ∈ R. We now assume that is a fixed but arbitrary derivation on R and that every derivation under consideration is a -derivation. Also, if N is a submodule of an R-module M, then for any x ∈ M, (N : x) will denote the right ideal of R given by {a ∈ R | xa ∈ N }.

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“…Colocalizations have been investigated under various approaches by several authors, for example see [3,5,12].…”

Section: Derivations and Modules Of Coquotientsmentioning

confidence: 99%

Differential torsion theory

Bland

2006

Journal of Pure and Applied Algebra

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“…We now show that a result similar to Proposition 4.2 holds for colocalizations of modules whenever they universally exist. Colocalizations have been investigated under various approaches by several authors, for example, see [3,6,13].…”

Section: Higher Derivations and Modules Of Coquotientsmentioning

confidence: 99%

Higher derivations on rings and modules

Bland

2005

International Journal of Mathematics and Mathematical Sciences

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Let ÃÂ„ be a hereditary torsion theory on ModR and suppose that QÃÂ„:ModRÃ¢Â†Â’ModR is the localization functor. It is shown that for all R-modules M, every higher derivation defined on M can be extended uniquely to a higher derivation defined on QÃÂ„(M) if and only if ÃÂ„ is a higher differential torsion theory. It is also shown that if ÃÂ„ is a TTF theory and CÃÂ„:MÃ¢Â†Â’M is the colocalization functor, then a higher derivation defined on M can be lifted uniquely to a higher derivation defined on CÃÂ„(M)

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Cotorsion theories

Cited by 2 publications

References 4 publications

Differential torsion theory

Differential torsion theory

Higher derivations on rings and modules

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