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.
Nearly every ring theory text says it is not. I bet guessing that 90% of texts agree with this is a conservative guess.
The main thing is what the first sentence you cited tries to say (but does not do a good job with): we want to say that $R/P$ is a prime ring if and only if $P$ is a prime ideal. (For noncommutative rings, this becomes $R/P$ is a domain if and only if $P$ is a completely prime ideal, that is, one satisfying the commutative definition of "prime." This does not matter much for the following discussion.)
If $\{0\}$ were considered a domain, then $R/R$ would be a domain for any ring $R$, so that $R$ is a prime ideal of itself. This is (in every reference I know) explicitly ruled out by the definition of a prime ideal. You could of course object that this is just an equivalent question ("why shouldn't the whole ring be considered a prime ideal?")
A very similar discussion took place here where someone asked why the zero ring wasn't considered a field. The argument that I gave there applies pretty well in this situation too. Our intuition of primes being "indecomposable" leads us to the idea that a prime ring shouldn't decompose into other rings. Certainly decomposing a prime ring into a product of multiple prime rings would be unattractive. But if you let $R=\{0\}$ be a prime ring, then $R\cong\prod_{i\in I} R$ or any nonempty index set that you like.
In particular if we let $\{0\}$ be called a domain, then we'd have a domain that is isomorphic to infinitely many copies of itself, which certainly seems like poor behavior for a domain.