In a different question that I asked (see here), there were examples of rings in which not every element is divisible by an irreducible, so my question is, do the rings which have the property of all nonzero nonunit elements being divisible by an irreducible have a specific name or classification?

Some common examples are $\mathbb{Z}$, $\mathbb{Z}[i]$, and most other standard integral domains, and technically any field counts because there are no nonzero non-unit elements.

  • $\begingroup$ I don't know, but in practice a useful sufficient condition is that every Noetherian integral domain has this property. $\endgroup$ – Qiaochu Yuan Mar 31 '15 at 4:34
  • $\begingroup$ Also, $\mathbb{N}$ is not a ring. $\endgroup$ – Qiaochu Yuan Mar 31 '15 at 4:37
  • $\begingroup$ Haha oops, I'll change that $\endgroup$ – ASKASK Mar 31 '15 at 4:38

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