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Math Help - Subrings

  1. #1
    Super Member Bernhard's Avatar
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    Subrings

    Find all positive integers n such that  \mathbb{Z}_n contains a subring isomorphic to  \mathbb{Z}_2
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  2. #2
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    Re: Subrings

    clearly, we must have a subgroup isomorphic to (Z2,+), so n must be even. so if n = 2k, then this subgrioup is {[0],[k]}.

    for this to be a subring of Zn isomorphic to Z2, we must have [k]*[k] = [k], that is:

    k2 = k (mod n).

    note that k2 = k + k +...+ k (k summands), if k is even (that is, n is divisible by 4) then this sum is 0 (mod n).

    hence k must be odd. that is, n must be of the form n = 4t-2, for some positive integer t.

    on the other hand, suppose n = 4t-2, and consider {[0],[2t-1]} in Zn. we have:

    [2t-1]*[2t-1] = [4t2-4t+1] = [4t2-2t] - [2t-1] = [4t-2]*[1] - [2t-1] = [0]*[1] - [2t-1] = -[2t-1] = -[2t-1] + [0] = -[2t-1] + [4t-2] = [2t-1].

    note this shows that even in a ring with identity, a subring may have a different identity than the parent ring (something that is NOT true of groups).

    some examples to consider: {[0],[3]} in Z6, and {[0],[15]} in Z30.
    Thanks from Bernhard
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