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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?

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  • $\begingroup$ I doubt that $\mathbb Z / 4 \mathbb Z$ is a domain. $\endgroup$ – user26857 Sep 11 '16 at 4:09
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A ring in which all ideals are principal, but which is not (necessarily) an integral domain is known as a principal ideal ring.

The combination of having both properties is just useful often enough to have its own combined name and abbreviation. In particular, the combination seems to allow us to transfer a lot more intuition and results from the integers than either of the properties by itself would have.

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  • $\begingroup$ Oh now I see what confused me: In german Hauptidealring ("principal ideal ring") is always used interchangeably with Hauptidealbereich ("principal ideal domain"), both with the condition of having no zero divisors. There seems to be no established term for principal ideal ring. $\endgroup$ – flawr Sep 9 '16 at 14:45
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A domain, by definition, has no zero-divisor. If we drop this assumption, we lose Euclidean division and unique factorisation into irreducible elements.

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