Maximal Ideal of Commutative and Unitary Ring is Prime Ideal

Theorem

Let $R$ be a commutative ring with unity.

Let $M$ be a maximal ideal of $R$.

Then $M$ is a prime ideal of $R$.

Proof

From Maximal Ideal iff Quotient Ring is Field:

the quotient ring $R / M$ is a field.

It follows from Field is Integral Domain that $R / M$ is an integral domain.

Finally it follows from Prime Ideal iff Quotient Ring is Integral Domain that $M$ is a prime ideal.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $13$