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HomA(M,HomA(N,K)) is isomorphic to HomA(N,HomA(M,K))

  1. Jan 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Let A be a commutative ring with identity element.

    Prove that HomA(M,HomA(N,K)) is isomorphic to HomA(N,HomA(M,K)).

    2. Relevant equations



    3. The attempt at a solution

    I believe it is best to start by defining a map, f: HomA(M,HomA(N,K) → HomA(N,HomA(M,K))
    for ψ: M → HomA(N,K) so that f(ψ)(n): m → ψ(m)(n).

    Then I guess I need to show this is an isomorphism of A-modules.

    However I'm not sure how to proceed. In similar questions I have defined another map usually in the opposite direction and shown they are inverse. This time though I'm not sure where to go.


    Any help would be appreciated.

    Thanks in advance!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 18, 2013 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    What are N,M,K?
     
  4. Jan 18, 2013 #3
    A-modules.
     
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