The notion of a domain 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).

 A: 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.



*

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

*"$R$ 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.

*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.
