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Basis of a module

  1. Mar 24, 2009 #1

    quasar987

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    Why couldn't a module have a finite basis and an infinite one?
     
  2. jcsd
  3. Mar 24, 2009 #2
    I think the reasoning goes something like this:

    Consider bases for some R-module, say A,
    [tex] \{x_{i}\}_{i=1}^{n} \mbox{ and } \{y_{i}\}_{i=1}^{\infty} [/tex]
    Everything in A can be written as a FINITE linear combination of elements from either basis. That means there is some natural number m with the property
    [tex] x_{i} = \sum_{j=1}^{m} \alpha_{i}^{j} y_{j}, \forall 1 \leq i \leq n \mbox{ and } \alpha_{i}^{j} \in R [/tex]
    Every x can be written as a linear combination of y's from some finite subset.

    Now consider an arbitrary element in A it can be written in terms of finite number of x's which can be written in terms of a finite number of y's thus everything can be written using a finite subset of the y's. In particular
    [tex] y_{m+1} = \sum_{i=1}^{n} \beta_{i}x_{j} = \sum_{i=1}^{n}\sum_{j=1}^{m} \beta_{i}\alpha_{i}^{j} y_{j} [/tex]

    This proves that the infinite basis is not linearly independent therefore in fact not a basis. If you find a flaw in this working or have more question tell me.
     
  4. Mar 24, 2009 #3

    quasar987

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    Oh, very nice. Thank you!
     
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