# Maximal implies prime

From Commalg

*This article gives the statement (and possibly proof) of an implication between properties of ideals in commutative unital rings; viz., any ideal in a commutative unital ring satisfying the first property, also satisfies the second*

## Contents

## Statement

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

## Definitions used

### Maximal ideal

`Further information: Maximal ideal`

### Prime ideal

`Further information: Prime ideal`

## Proof

### 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.

### Hands-on proof

*Fill this in later*