Is the zero ring usually considered a domain or not? Wikipedia says:
- The zero ring is not an integral domain; this agrees with the fact that its zero ideal is not prime. Whether the zero ring is considered to be a domain at all is a matter of convention, but there are two advantages to considering it not to be a domain. First, this agrees with the definition that a domain is a ring in which $0$ is the only zero divisor (in particular, $0$ is required to be a zero divisor, which fails in the zero ring). Second, this way, for a positive integer $n$, the ring $\Bbb Z/n\Bbb Z$ is a domain if and only if n is prime.
What are the arguments for/against this convention (besides the ones listed above)? What does the literature (i.e. your favorite textbooks) say on this matter? Note that I am specifically talking about domains, not integral domains; the only difference between them aside from the non-triviality assumption mentioned here is that domains are not required to be commutative.