Zero ideal and domains

Question

Let $R \neq 0$ be a commutative ring. I think we have

$$ R \text{ is a domain } \iff (0) \text{ is a prime ideal of } R.$$

The argument is straightforward: let $a,b \in R$ such that $a\cdot b = 0$. Then to say that $(0)$ is prime is equivalent to say that at least one of $a$ or $b$ is zero, i.e. that $R$ is an integral domain.

Do you agree with this or am I missing some possible issues?

You should prove that $P\subset R$ is a prime ideal if and only if $R/P$ is an integral domain. Then your statement is the case of $P=0.$ — Ragib Zaman, Dec 01 '13 at 12:50
@MartinBrandenburg Oh yes it's true we can also deduce the general statement from the $P=0$ case quickly as well. I'm just biased because I usually think of OP's statement as a corollary of the general statement rather than conversely. — Ragib Zaman, Dec 01 '13 at 13:00
@Ragib This answer has more on factoring out primes to reduce to domains, including quotes from a master algebraist (Kaplansky). Read the masters! — Bill Dubuque, May 20 '14 at 00:12

score 1 · Answer 1 · answered Feb 26 '15 at 03:10

From what I can see you only gave one implication: If $(0)$ is a prime ideal, then $R$ is a domain.

Conversely, if $R$ is a domain and $ab \in (0)$, then $ab = 0$ and thus by definition of integral domain, either $a = 0$ which means $a \in (0)$ or $b = 0$ which means $b \in (0)$. Conclude by definition that $(0)$ is a prime ideal.

score 1 · Answer 2 · answered Dec 01 '13 at 14:09

1

A much nicer proof (due to Hecke) is that $R/0\cong R$. $\square$

answered Dec 01 '13 at 14:09

Zero ideal and domains

2 Answers2