An ideal is prime iff is an integral domain (that is, it contains no nontrivial zero divisors). If is a prime ideal such that , then in the quotient ring the elements of the form , where , will be nilpotents. Thus, is not a domain and is not prime. For a concrete and illustrative example, consider - the ideal of the ring of integers generated by a prime . Then, , and is not prime (since it is generated by a non-prime integer).