Maximal implies prime

Statement
Any maximal ideal in a commutative unital ring is a prime ideal.

As quotient-determined properties
One proof uses the characterization of maximal and prime ideals in terms of their quotients, namely:


 * An ideal is maximal iff the quotient ring is a field
 * An ideal is prime iff the quotient ring is an integral domain

We know that the property of commutative unital rings of being a field is stronger than the property of being an integral domain, hence any maximal ideal is prime.