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Math Help - Maximal Ideal

  1. #1
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    Maximal Ideal

    Let R = Z x Z = {(x,y)|x,y in Z}

    Show that Z x 2Z is maximal in R. Where, Z x 2Z = {(x,y)| x in Z, y in 2Z}
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  2. #2
    Super Member Gamma's Avatar
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    quotient out by it you get \mathbb{Z}_2 which is a field, thus that ideal is maximal.
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  3. #3
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    I know that Z/2Z = Z_2 (is a field) but how do you know that ZxZ/Zx2Z is a field?
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  4. #4
    Super Member Gamma's Avatar
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    Just take a look at the cosets. (a,b)+(Z, 2Z)=(a+Z,b+2Z)=(Z,b+2Z). There are only two cases, if b is even, it is the 0, if b is odd, it is the other one. Thus it is a group with 2 elements, it is the finite field of order 2.
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