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    Ideals and Factor Rings : Prime Ideal

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    Problem:
    Let R be a commutative ring.
    Show that every maximal ideal of R is prime and if R is finite, show that every prime ideal is maximal.
    ALSO is every prime ideal of Z(integers) maximal? Why?

    © BrainMass Inc. brainmass.com February 24, 2021, 2:23 pm ad1c9bdddf
    https://brainmass.com/math/ring-theory/ideals-factor-rings-prime-ideal-17287

    Solution Preview

    Proof:
    R is a commutative ring. Then M is an maximal ideal of R if and only if R/M is a field, P is a prime ideal if and only if R/P is an integral domain.
    (1) If M is an maximal ideal, then R/M is a field. So if xy is in M, we consider x+M and y+M in R/M. Since R/M is a field, we have ...

    Solution Summary

    A proof involving a prime ideal is provided in the solution.

    $2.19

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