Question

Show that a quotient of an integral domain need not be an integral domain.

Answer 1

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/4Z is not an integral domain.
In fact, the element 2+4Z is a nonzero element in Z/4Z.

However, the product,

(2+4Z)(2+4Z)=4+Z=0+Z is a zero in Z/4Z.

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

Note : 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=4Z is not a prime ideal of Z.