# Is the Quotient Ring of an Integral Domain still an Integral Domain?

## Problem 589

Let $R$ be an integral domain and let $I$ be an ideal of $R$.

Is the quotient ring $R/I$ an integral domain?

## Definition (Integral Domain).

Let $R$ be a commutative ring.

An element $a$ in $R$ is called a **zero divisor** if there exists $b\neq 0$ in $R$ such that $ab=0$.

If $R$ contain no nonzero zero divisors, then $R$ is called an **integral domain**.

## Solution.

The quotient ring $R/I$ of an integral domain is not necessarily an integral domain.

Consider, for example, the ring of integers $\Z$ and ideal $I=4Z$.

Note that $\Z$ is an integral domain.

We claim that the quotient ring $\Z/4\Z$ is not an integral domain.

In fact, the element $2+4\Z$ is a nonzero element in $\Z/4\Z$.

However, the product

\[(2+4\Z)(2+4\Z)=4+\Z=0+\Z\]
is zero in $\Z/4\Z$.

This implies that $2+4\Z$ is a zero divisor, and thus $\Z/4\Z$ is not an integral domain.

## Comment.

Note that in general, the quotient $R/I$ is an integral domain if and only if $I$ is a prime ideal of $R$.

In our above example, the ideal $I=4\Z$ is not a prime ideal of $\Z$.

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