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Math Help - Ring Isomorphism

  1. #1
    Super Member Bernhard's Avatar
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    Ring Isomorphism

    Can anyone help with the following problem:

    Prove that the rings 2\mathbb{Z} and 3\mathbb{Z} are not isomorphic?

    Be grateful for help

    Peter
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  2. #2
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    Re: Ring Isomorphism

    How many members does 2\mathbb{Z}? What about 3\mathbb{Z}?
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Ring Isomorphism

    2Z = { ... -4.-2, 0, 2, 4, 6, ....} so it is infinite

    3Z = { ... -6. -3, 0, 3, 6, .... } so it is infinite

    ... or have I got something wrong?

    Peter
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  4. #4
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    Re: Ring Isomorphism

    Quote Originally Posted by Bernhard View Post
    Can anyone help with the following problem:

    Prove that the rings 2\mathbb{Z} and 3\mathbb{Z} are not isomorphic?

    Be grateful for help

    Peter

    Let f:2\mathbb{Z} \to 3\mathbb{Z} be a ring isomorphism. Then f(2) \in 3\mathbb{Z}. Thus f(2) = 3 x, x \in \mathbb{Z}.

    Therefore f(4) = f(2+2) = f(2)+f(2) = 6x and f(4)=f(2\cdot2)=f(2) f(2)=9x^2.

    So, 6x=9x^2 \Rightarrow x=0 or x=2/3.

    x=0 means f(2)=0, so f(2\lambda ) = f(2)f(\lambda )=0, \forall \lambda \in \mathbb{Z} which means that f is not an isomorphism,

    and x=2/3 with x\in \mathbb{Z} is Absurd.

    So, the rings 2\mathbb{Z} and 3\mathbb{Z} are not isomorphic.
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  5. #5
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    Re: Ring Isomorphism

    Sorry, I was thinking " \mathbb{Z} modulo 2 and 3". zoek has it right.
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  6. #6
    Super Member Bernhard's Avatar
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    Re: Ring Isomorphism

    No problems. Thanks anyway

    And thanks for help with previous posts as well!

    Peter
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