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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.

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  • $\begingroup$ I can't think of any argument for it. On the other hand, excluding the $0$ ring in the definition means to not have to consider it as a special case or exception in your proofs. $\endgroup$
    – A.P.
    Jun 15, 2015 at 17:57
  • $\begingroup$ People often exclude the zero ring even as a ring (i.e. By prefacing a statement with "Let $R$ be a ring with $0\ne 1$..."). It's pretty convenient for an ring to have a quotient which is a field. If someone wants to exclude/include the zero ring into integral domains, I'd like to know why they'd include it into rings. $\endgroup$
    – PVAL-inactive
    Jun 15, 2015 at 18:03
  • $\begingroup$ @A.P. There's no free lunch. If you make extra assumptions in your proof, that just means that it won't be as widely applicable, and users of the theorem will need to work harder to prove the extra assumptions, even if they are not necessary. I find it decidedly unnatural to go so far as to exclude the zero ring as a ring, because it has all the properties you would expect of a ring. The issue comes in when you start dividing things and $0$ is invertible, but domains are on the border of acceptability here. $\endgroup$
    – Mario Carneiro
    Jun 15, 2015 at 18:32
  • $\begingroup$ Sure, someone would need to prove the result for the $0$ ring. The point is that most results are either trivial or inapplicable for the $0$ ring, so having to mention it every time is just an annoyance. $\endgroup$
    – A.P.
    Jun 15, 2015 at 18:49

2 Answers 2

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Is the zero ring usually considered a domain or not?

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.)

What does the literature (i.e. your favorite textbooks) say on this matter?

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.

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  • $\begingroup$ Regarding "Why shouldn't the whole ring be a prime ideal?", for some reason I am much more comfortable with this definition. Similar to maximal ideals or filters in topology ($\emptyset\notin F$), without this assumption many statements tend to trivialize without adding the condition back in anyway. $\endgroup$
    – Mario Carneiro
    Jun 15, 2015 at 19:31
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I would like domains to have fields of fractions. Fractions are equivalence classes of ordered pairs $(p,q)$ where $q \neq 0$. Since all elements of the zero ring are zero, there are no such pairs and one deduces that the "field of fractions" should be the empty set. But the empty set is not a field (does not contain $0$ or $1$.)

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  • $\begingroup$ I don't buy this argument. Of course you aren't going to get a field of fractions, because the multiplication isn't commutative. But I'm not even sure you will get a well-defined skew field / division ring with the usual construction. Even the total ring of fractions, which allows zero divisors, has to assume that the ring is commutative at the outset. $\endgroup$
    – Mario Carneiro
    Jun 15, 2015 at 18:09
  • $\begingroup$ See also: Ore condition, which details the conditions under which one can expect a noncommutative ring or domain to have an associated division ring of fractions. It would seem that zero ring cannot be an Ore domain for the results on that page to hold. $\endgroup$
    – Mario Carneiro
    Jun 15, 2015 at 18:22
  • $\begingroup$ @MarioCarneiro What do you mean the zero ring isn't commutative? $0 \times 0 = 0 = 0 \times 0$. $\endgroup$
    – David E Speyer
    Jun 15, 2015 at 19:26
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    $\begingroup$ @Mario: It is common in many parts of mathematics to define a domain to be a commutative ring without zero divisors. In particular, many people who study commutative algebra do not like the term integral domain, because the word "integral" has a completely different meaning in that subject. (David: your proof of the commutativity of the zero ring can be simplified!) $\endgroup$
    – Pete L. Clark
    Jun 15, 2015 at 21:08
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    $\begingroup$ (I checked out the project. It looks like you folks are doing very nice work.) $\endgroup$
    – Pete L. Clark
    Jun 16, 2015 at 2:38

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