# Section16.4Maximal and Prime Ideals¶ permalink

In this particular section we are especially interested in certain ideals of commutative rings. These ideals give us special types of factor rings. More specifically, we would like to characterize those ideals $I$ of a commutative ring $R$ such that $R/I$ is an integral domain or a field.

A proper ideal $M$ of a ring $R$ is a maximal ideal of $R$ if the ideal $M$ is not a proper subset of any ideal of $R$ except $R$ itself. That is, $M$ is a maximal ideal if for any ideal $I$ properly containing $M$, $I = R$. The following theorem completely characterizes maximal ideals for commutative rings with identity in terms of their corresponding factor rings.

##### Example16.36

Let $p{\mathbb Z}$ be an ideal in ${\mathbb Z}$, where $p$ is prime. Then $p{\mathbb Z}$ is a maximal ideal since ${\mathbb Z}/ p {\mathbb Z} \cong {\mathbb Z}_p$ is a field.

A proper ideal $P$ in a commutative ring $R$ is called a prime ideal if whenever $ab \in P$, then either $a \in P$ or $b \in P$. 5

##### Example16.37

It is easy to check that the set $P = \{ 0, 2, 4, 6, 8, 10 \}$ is an ideal in ${\mathbb Z}_{12}$. This ideal is prime. In fact, it is a maximal ideal.

##### Example16.39

Every ideal in ${\mathbb Z}$ is of the form $n {\mathbb Z}$. The factor ring ${\mathbb Z} / n{\mathbb Z} \cong {\mathbb Z}_n$ is an integral domain only when $n$ is prime. It is actually a field. Hence, the nonzero prime ideals in ${\mathbb Z}$ are the ideals $p{\mathbb Z}$, where $p$ is prime. This example really justifies the use of the word “prime” in our definition of prime ideals.

Since every field is an integral domain, we have the following corollary.