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Math Help - Boolean commutative ring

  1. #1
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    Boolean commutative ring

    I'm just starting work on Boolean rings. I understand so far that if we have some commutative Boolean ring $ R$ with unity, that for every \alpha\in R we have \alpha^2 = \alpha.

    I would think then that some ideal I in R would make the factor ring R/I also boolean. How can I prove that this factor ring R/I is also boolean?

    As for maximals, how is it we would prove that every proper nontrivial prime ideal I<R is maximal?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by DanielThrice View Post
    I'm just starting work on Boolean rings. I understand so far that if we have some commutative Boolean ring $ R$ with unity, that for every \alpha\in R we have \alpha^2 = \alpha.

    I would think then that some ideal I in R would make the factor ring R/I also boolean. How can I prove that this factor ring R/I is also boolean?

    As for maximals, how is it we would prove that every proper nontrivial prime ideal I<R is maximal?
    These questions seem to be coming from overarching question, did you forget to include it?
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  3. #3
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    if R is boolean, then for any quotient R/I, we have: (α + I)(α + I) = α^2 + I = α + I, so R/I is boolean.

    suppose I is a proper non-trivial prime ideal. then ab in I implies either a in I, or b in I, so R/I is an integral domain. but R/I is also boolean, so if a ≠ 0 in R/I,

    then a = a^2 = a^3. thus a(1) = a(a^2), so a^2 = 1. hence a = 1 or -1. but -1 = (-1^2) = 1, so R/I = {0,1}.

    from distributivity we have (1 + 1)(1 + 1) = 1 + 1 + 1 + 1, from the fact that R/I is boolean, we have (1 + 1)(1 + 1) = 1 + 1.

    thus 1 + 1 = 0, so R/I is isomorphic to F2. since F2 is a field, I must be maximal.
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  4. #4
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    By F2 do you mean \mathbb Z_2 or just some generic field?

    Thank you, that's a pretty clear way of looking at it
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  5. #5
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    Quote Originally Posted by Drexel28 View Post
    These questions seem to be coming from overarching question, did you forget to include it?
    And no this is a basic theory kinda problem, no specifics
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  6. #6
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    F2 is the finite field of 2 elements. Z2 is another name for it, yes.
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