In the usual definition, a principal ideal domain $R$ is also assumed to be an integral domain. However, the property that every ideal is generated by a single element does not seem to immediately imply that the ring is integral. Is this correct and if so:
Do there exist rings where every ideal is generated by a single element and has zero divisors?
I am most interested in the case where $R$ is commutative with unity, but don't mind examples where these properties don't hold.
Also, assuming there are examples, is there any reason why we make this assumption?