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

Problems and solutions of ring theory in abstract algebra

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?

 
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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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