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Ideals and Factor Rings : Prime Ideal

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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?

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Solution Summary

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

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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 ...

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