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Math Help - Rings

  1. #1
    Junior Member
    Joined
    Jan 2009
    Posts
    64

    Rings

    Hi, The following is a challenge question that was set, which i do not know how to do;


    Rings-123.jpg

    The following is the defintion that was given to us:

    Rings-untitled.jpg

    Can anyone do this problem.
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  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Hi

    Let m=2^ap_1^{a_1}...p_r^{a_r} a positive integer, p_1,...,p_r distincts even primes. The group of units of the ring (\mathbb{Z}_m,+,\times) is isomorphic to (G\times\prod\limits_{i=1}^{r}\mathbb{Z}_{\phi(p_i  ^{a_i})},+) where \phi(p_i^{a_i})=p_i^{a_i-1}(p_i-1) and G is:

    \{0\} if a=0,1
    \mathbb{Z}_2 if a=2
    \mathbb{Z}_2\times\mathbb{Z}_{2^{a-2}} if a\geq 3

    If I'm not wrong, that should be sufficient to prove that m=2,3,4,6,8,12,24 are the only solutions; but I let you show this, or disprove it
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  3. #3
    Junior Member
    Joined
    Jan 2009
    Posts
    64
    Thanks for the help;

    I should be able to slove this problem know.
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