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)


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 \( \mathcal{C} \).


Direct Summand Cardinal Number Unitary Ring Regular Cardinal Decomposition Property 
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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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