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We can’t rely on global conventions on the nLab. If there is a terminological ambiguity, best to say so explicitly in any entry which it effects.
Historically, integral domain is an abbreviation of “domain of integrality” and domain is an abbreviation of “integral domain”. Most commutative algebraists by ring mean commutative ring and by integral domain commutative integral domain. Most noncommutative ring theorists, at the time when such specialists existed (I knew quite a few of them including one of my professors Levy from whom I took a comprehensive ring theory course – and most of them are now either retired and inactive or died) say interchangeably domain and integral domain in noncommutative context and most of the people in commutative algebraic geometry, number theory and commutative algebra have commutative convention for rings and domains. In some other languages it is still “domain of integrality” in either context, not abbreviated (in Russian: oblast’ celostnosti). I like to say integral domain in geometric context when first mentioning so that no confusion/association appears e.g. with a domain in the sense of topology or a domain in the sense of geometry, or a domain in the sense of complex analysis. But in ring theory both terms are perfectly good and interchangeable and context has to be anyway specified beforehand.
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