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Math Help - Isomorphisms and rings

  1. #1
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    Isomorphisms and rings

    If R is a ring with a in R, then the set Ra={ra such that r is in R}, then Ra is a subring of R.

    Set S={0,3,6,9,12,15,18,21} and find an isomorphism f:Z(mod 8)->S to show S and Z(mod 8) are isomorphic.

    Set T={0,6,12,18}. Is T isomorphic to Z(mod 4)? Explain.


    I'm having trouble coming up with the isomorphism for the first part. Any ideas what the function might be? I was thinking it could just easily be set piecewise, that is f(1)=9, f(2)=18 and so on, but is that what I would need to do, or is there a more intricate solution?
    I think for the second part, it is no because 0 is 0 mod 4, 6 is 2 mod 4, 12 is 0 mod 4, and 18 is 2 mod 4...so we could not make a function f:Z(mod 4)->T that is one-to-one. Is this correct?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zhupolongjoe View Post
    If R is a ring with a in R, then the set Ra={ra such that r is in R}, then Ra is a subring of R.

    Set S={0,3,6,9,12,15,18,21} and find an isomorphism f:Z(mod 8)->S to show S and Z(mod 8) are isomorphic.

    Set T={0,6,12,18}. Is T isomorphic to Z(mod 4)? Explain.


    I'm having trouble coming up with the isomorphism for the first part. Any ideas what the function might be? I was thinking it could just easily be set piecewise, that is f(1)=9, f(2)=18 and so on, but is that what I would need to do, or is there a more intricate solution?
    I think for the second part, it is no because 0 is 0 mod 4, 6 is 2 mod 4, 12 is 0 mod 4, and 18 is 2 mod 4...so we could not make a function f:Z(mod 4)->T that is one-to-one. Is this correct?
    What structure do S and T have?
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    What structure do S and T have?
    Oh, I apologize. They are subrings of Z24.
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