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Math Help - Isomorphism in module

  1. #1
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    Isomorphism in module

    Let A,B be R-modules, and let f: A -> B be an onto module homomorphism and let C be the kernel of f. Prove that A/C is isomorphic.

    Proof so far:

    Let g : A/C -> B be defined by g(a+C) = f(a)

    My claim is that g is an isomorphism.

    Well-defined: Pick an element in A/C, say x, then x is in A but not in C. meaning f(x)  \neq 0, since f is well-defined, g also is well defined.

    Linear: Let x,y be in A/C, and let s be a scalar. Again, f(x) and f(y) are not equal to 0.

    g(x+C) = f(x) = u , g(y+C) = f(y) = v , for elements u,v in B.

    g(x+y+C) = f(x+y) = f(x) + f(y) = u + v since f is an homomorphism.

    Let s be a scalar, g(s(x+C)) = f(sx) = sf(x) since f is module homomorphism.

    So g is also linear.

    Am I doing this right so far?
    Last edited by tttcomrader; March 27th 2008 at 07:18 PM.
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  2. #2
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    The following up problem is related to this previous one:

    Let M be an R-module, and let L,N be submodules. Show that L/(L \cap N ) \cong (L+N)/N

    proof so far:

    Define  p: L \rightarrow (L + N)/N by  p(x) = x + N

    My goal is to prove p is an onto homomorphism, then use the previous exercise.

    Will this be a right start?
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