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The notion of an integral domain originally included the following properties (https://en.wikipedia.org/wiki/Integral_domain):

  • it is nonzero;
  • it does not have nonzero zero divisors;
  • it has identity;
  • it is commutative.

Later, the notion of a domain has been widened to include non-commutative rings and/or rngs
(https://en.wikipedia.org/wiki/Domain_(ring_theory)):

  • it is nonzero;
  • it does not have nonzero zero divisors;
  • it has identity;

or

  • it is nonzero;
  • it does not have nonzero zero divisors;
  • it is commutative.

The common denominator of all the definitions are the two properties:

  • it is nonzero;
  • it does not have nonzero zero divisors.

It looks like we are still missing the name for one the defining properties:
the non-existence of nonzero zero divisors. If I understand it correctly, the word "integral" in the original term had exactly this meaning.

Is it acceptable to call a ring/rng with no nonzero zero divisors an integral ring/rng?

In this case we could define a domain as a ring/rng with the following properties:

  • it is nonzero (mandatory);
  • it is integral (mandatory);
  • it has identity (optional);
  • it is commutative (optional).
J. W. Tanner
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Alex C
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    The word integral has also the following meaning. I don't like too many "integrals". Integral domain and integral closure is enough. – Dietrich Burde Feb 25 '20 at 16:26
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    Rings without nonzero zero divisors are also called cancellative (just like for semigroups and monoids). The minor variations in convention are widespread so there is little hope of establishing any standard. – Bill Dubuque Feb 25 '20 at 16:56
  • @BillDubuque I like cancellative. Thank you. – Alex C Feb 25 '20 at 17:00

1 Answers1

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The observations you have made so far are astute. Let me give my following report on what I think the common usage is, addressing the question asked in passing:

Is it acceptable to call a ring/rng with no nonzero zero divisors an integral ring/rng?

You are right that the one condition that cannot be omitted is the one you call "being integral." I think everyone agrees that condition is essential. But this deserves a comment:

the non-existence of nonzero zero divisors. If I understand it correctly, the word "integral" in the original term had exactly this meaning.

The whole question of why "integral" is even used to describe such rings is one I can't do justice to, but this previous excellent post does a good job. I think the two terms "integral domain" evolved together to talk about the absent zero divisors, and maybe separating them is an artificial thing to do. You can call a domain an integral ring but I don't think it's the best choice.

  1. "$R$ is an integral domain" will be almost always interpreted as commutative and having identity.

  2. "$R$ is a domain" will often be interpreted the same as "integral domain," but noncommutative algebraists are accustomed to omitting commutativity from it and probably always assume identity as well.

  3. In my opinion, it is exceedingly rare for a ring without nonzero zero divisors and without an identity to be called a domain. This is not often a subject of study, and shouldn't be the thing you think of when you see "domain."

There is one noteworthy edge case: $\{0\}$ lacks nonzero zero divisors and has an identity, but nobody calls it a domain.

rschwieb
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