Maximal Ideal of Commutative and Unitary Ring is Prime Ideal
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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 $13$