Maximal Ideal of Commutative and Unitary Ring is Prime Ideal

Theorem
Let $$A$$ be a commutative ring with unity.

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

Then $$M$$ is prime.

Proof
It follows from Maximal Ideal iff Quotient Ring is Field that 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 in Integral Domain that $$M$$ is a prime ideal.