# Why do we require a principal ideal domain to be an integral domain?

I just noticed that all definitions of principal ideal domains exclude the possibility of zero divisors. But why is that?

If we drop this requirement we could say that e.g. $\mathbb Z / 4 \mathbb Z$ is a principal ideal domain. What backdraws are there or what properties would we lose if we dropped the requirement of being an integral domain?

• I doubt that $\mathbb Z / 4 \mathbb Z$ is a domain. – user26857 Sep 11 '16 at 4:09