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Why are integral domains required to be non-zero? I understand this is mostly a convention, but it seems somewhat arbitrary to me as of now. Is there any known theorem that fails if we allow the $0$ ring to be considered an integral domain? Or is it because of purely historical reason? I'm wondering the same for, PID, fields, etc; but since they are supposed to be a subset of integral domains, then the corresponding question reduces to the question of integral domains.

Fernando Chu
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    Well, for instance, it often leads to some trivial counterexamples that no one really wants to consider. The zero ring just doesn't behave in the same way many other domains/fields/whatever do. While focused on fields, you can find some discussion here – PrincessEev Nov 15 '21 at 22:33
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    I like to be able to say that, in an integral domain, any finite product of non-zero elements is non-zero. If the zero ring were called an integral domain, then the empty product would be a counterexample. – Andreas Blass Nov 16 '21 at 01:36

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