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On the Construction of Separable Modules

  • P. A. Guil Asensio
  • M. C. Izurdiaga
  • B. Torrecillas
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Let R be an associative ring with identity and κ, an infinite regular cardinal. A left R-module M is said to be κ -generated (resp. κ <-generated) if there exist a generator set {m i } I of M of cardinality at most κ (resp. strictly smaller than κ And a module M is called κ-separable if any subset X of M of cardinality strictly smaller than κ is contained in a κ <-generated direct summand of M. Let us no that any direct sum of κ <-generated modules is clearly a κ-separable module that we will call trivial. This notion of separability can be extended as follows. Given an infinite regular cardinal κ and a non-empty class of modules \( \mathcal{C} \), we may say that a module M is (κ, \( \mathcal{C} \))-separable if each subset X of M of cardinality strictly smaller than κ is contained in a direct summand of M belonging to \( \mathcal{C} \). Again, we will say that the (κ, \( \mathcal{C} \))-separable module M is non-trivial if it is not a direct sum of elements in \( \mathcal{C} \). These modules present a pathological behavior in the sense that they have enough direct summands belonging to \( \mathcal{C} \), but they are not direct sums of modules in \( \mathcal{C} \).

Keywords

Direct Summand Cardinal Number Unitary Ring Regular Cardinal Decomposition Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • P. A. Guil Asensio
    • 1
  • M. C. Izurdiaga
    • 2
  • B. Torrecillas
    • 2
  1. 1.Departamento de MatemáticasUniversidad de MurciaEspinardo, MurciaSpain
  2. 2.Departamento de Álgebra y Análisis MatemáticoUniversidad de AlmeríaAlmeríaSpain

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