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Math Help - Show an isomorphism

  1. #1
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    Show an isomorphism

    I have to find this isomorphism:

    (P_{\alpha +1}\oplus Q_{\alpha +1})/(P_\alpha \oplus Q_\alpha) \simeq P_{\alpha +1}/P_\alpha \oplus Q_{\alpha +1}/Q_\alpha

    Where P_\alpha = S_\alpha \cap P and Q_\alpha = S_\alpha \cap Q and {S_\alpha} is a well ordered increasing sequence of sub modules for M=P\oplus Q.

    I know that it is something about sending the residue class (\overline{P_{\alpha +1}, Q_{\alpha +1}}) into the pair of residue classes (\overline{P_{\alpha +1}}, \overline{Q_{\alpha +1}})

    But how to argue that this is a homomorphism?
    And what would the kernel be?

    Any help would be nice, thanks.
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  2. #2
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    Quote Originally Posted by Carl View Post
    I have to find this isomorphism:

    (P_{\alpha +1}\oplus Q_{\alpha +1})/(P_\alpha \oplus Q_\alpha) \simeq P_{\alpha +1}/P_\alpha \oplus Q_{\alpha +1}/Q_\alpha

    Where P_\alpha = S_\alpha \cap P and Q_\alpha = S_\alpha \cap Q and {S_\alpha} is a well ordered increasing sequence of sub modules for M=P\oplus Q.


    What do you mean by "well-ordered increasing seq. of submod. of..."?? That is indexed by a countable index set and thus S_1\subset S_2\subset ....\subset S_n\subset\ldots , or

    only that for any two indexes \alpha,\,\beta , either S_\alpha\subset S_\beta\,\,\,or\,\,\,S_\beta\subset S_\alpha ?

    Tonio


    I know that it is something about sending the residue class (\overline{P_{\alpha +1}, Q_{\alpha +1}}) into the pair of residue classes (\overline{P_{\alpha +1}}, \overline{Q_{\alpha +1}})

    But how to argue that this is a homomorphism?
    And what would the kernel be?

    Any help would be nice, thanks.
    .
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  3. #3
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    Pick any (x,y)\in P_{\alpha+1}\oplus Q_{\alpha+1}, then your map is just (\overline{x,y})\mapsto(\bar{x},\bar{y}). You need to show that 1) the map is well-defined, by showing if (x1,y1), (x2,y2) differ by something in P_\alpha\oplus Q_\alpha, then they map to the same thing, and 2) is a homomorphism (additive property and scalar multiplication property), which is easy to prove, and 3) 1-1 and onto, which is also not hard.
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