Principal Ideal Ring which is not Integral In Atiyah & McDonald: Commutative Algebra the Principal Ideal Domain is a principal ideal ring which is also an integral domain.
I tried but couldn't find examples of commutative rings with identity that have the property that every ideal is generated by a single element but are not integral. Any suggestions ?
I believe that the fact that the set of all zero-divisors is not closed under "$+$" (in $\mathbb{Z_6}$ for example $3+2=5$) is making the search of such example difficult.

In general, if $R$ is a PID, then every quotient of $R$ is a PIR.
Ironically, $\Bbb Z_6$ is such a ring, because it is $\Bbb Z / (6)$.
