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Math Help - 2 Ring Isomorphism Questions

  1. #1
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    2 Ring Isomorphism Questions

    1.) Determine all ring isomorphisms from Zn to itself.

    2.) Determine all ring homomorphisms from Z+Z into Z+Z (where + is the Direct Product).

    For #1, I know that Zn has unity 1, so phi(1) = 1. Does that mean Phi(x) = x is the only ring isomorphism from Zn to itself?

    I'm really stuck on #2. Any help would be greatly appreciated.

    Thanks in advance!
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  2. #2
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    I may have gotten #2. Are there 9 different ring homomorphisms that work? Thanks.
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  3. #3
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    Quote Originally Posted by GB89 View Post
    For #1, I know that Zn has unity 1, so phi(1) = 1. Does that mean Phi(x) = x is the only ring isomorphism from Zn to itself?
    If \phi(1) = 1 then \phi(n) = \phi(1+...+1) = \phi(1) + ... + \phi(1) = n\phi(1) = n.

    Hint: If you know what \phi(1,0) and \phi(0,1) is then you can determine everything.
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