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In my algebra course (taught from Artin), a principal ideal domain is defined as an integral domain such that all ideals are principal. This got me wondering:

Are there rings for which every ideal is principal, but the ring is not an integral domain?

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    This was in the related questions column (and probably in the list of possible duplicates if you used the web interface to submit the question.) Please do a search first. – 2017-02-17
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    Also related: http://math.stackexchange.com/q/1920478/29335 – 2017-02-17

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Definitely! In fact there's a Wikipedia page about principal ideal rings.

The smallest example that is not a domain would be $\mathbb{Z}/4\mathbb{Z}$. More generally, you can take any principal ideal domain $R$ you know, and any ideal $I\subset R$ that is not a prime ideal, and then $R/I$ will be a principal ideal ring that is not a domain.