# 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?

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.

$2.49