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    Principal and Prime Ideals

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    Let R be a commutative ring with 1. Show that the principal ideal generated by x in the polynomial ring R[x] is a prime ideal if and only if R is an integral domain. Prove that (x) is a maximal ideal if and only if R is a field.

    I have a solution for this problem but it starts with:First, note that a polynomial f in R[x] belongs to the ideal (x) if and only if f (0) = 0.

    Can someone please clarify this note a bit.

    © BrainMass Inc. brainmass.com May 20, 2020, 5:30 pm ad1c9bdddf
    https://brainmass.com/math/integrals/principal-prime-ideals-227852

    Solution Preview

    Proof:
    "=>": If f in R[x] belongs to the ideal (x), then f(x) = xg(x) for some g(x) in R[x], then ...

    Solution Summary

    The expert examines principals and prime ideals generated by a polynomial ring.

    $2.19

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