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Math Help - R/M and R

  1. #1
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    R/M and R

    if M is the maximal ideal of R....how can we say that R/M has only 2 ideals ....0 and R/M itself??
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  2. #2
    Senior Member roninpro's Avatar
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    If R is a commutative ring with identity, and M is a maximal ideal, then R/M is a field. What are the ideals of a field?
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by roninpro View Post
    If R is a commutative ring with identity, and M is a maximal ideal, then R/M is a field. What are the ideals of a field?
    Or, I belive you could use the correspondence theorem.
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  4. #4
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    that was what i had set out to prove....that if If R is a commutative ring with identity, and M is a maximal ideal, then R/M is a field.....they directly said that R/M has 2 ideals 0 and R/M itself....so i cld'nt get how they said this...

    could you please tell me somethng abt this correspondence theorem. cz i dont seem to get what it says....
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by prashantgolu View Post
    that was what i had set out to prove....that if If R is a commutative ring with identity, and M is a maximal ideal, then R/M is a field.....they directly said that R/M has 2 ideals 0 and R/M itself....so i cld'nt get how they said this...

    could you please tell me somethng abt this correspondence theorem. cz i dont seem to get what it says....
    The correspondence theorem (I believe) basically says two things,

    1. Let I, J \lhd R with I \subseteq J. Then J/I \lhd R/I.

    2. If J/I \lhd R/I then there exists an ideal J^{\prime} \lhd R such that I \subseteq J^{\prime} and J^{\prime}/I = J/I.

    So, essentially, the ideals in R/I correspond precisely to the ideals J such that I \lhd J \lhd R. As there are no such proper ideals, as I is maximal, then what can you conclude?
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