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?

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    $\begingroup$ What about $\Bbb Z_4$? $\endgroup$ – Berci Jul 8 '15 at 10:10
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    $\begingroup$ Principal ideal rings are of course studied. The minimal example is $\mathbb{Z}/4\mathbb{Z}$. $\endgroup$ – egreg Jul 8 '15 at 10:11

Yes. Such rings are called principal ideal rings. An example of such a ring would be $K[x]/(x^2)$, where $K$ is any field.

In fact, a theorem of Hungerford states that any principal ideal ring is the direct product of quotients of principal ideal domains.


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