Question about the zero ideal in $\mathbb{Z}$

Question

I have been studying Ring theory and a question came up.

Theorem: A commutative ring $R$ is integral domain iff the zero ideal $I=\{0\}$ is prime ideal in $R$.

Question: How can it be that in $\mathbb{Z}$ we do not count $I=\{0\}$ as a prime ideal ?(That's what is stated in my book)

We know that $\mathbb{Z}$ is integral domain. So the zero ideal must be prime.

$(0)$ is a prime ideal in $\Bbb Z$, if $ab\in (0)$ then either $a\in (0)$ or $b\in (0)$. — cansomeonehelpmeout, Dec 23 '18 at 19:21
The zero ideal is a prime one iff the (non-trivial) ring is an integral domain. If it is stated otherwise in your book or anywhere then that is wrong — DonAntonio, Dec 23 '18 at 19:24
See Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?. If this is really what you are asking about then we can close as a dupe of that (and likely many others). — Bill Dubuque, Dec 23 '18 at 19:36

score 2 · Answer 1 · answered Dec 23 '18 at 19:22

2

An ideal is prime if whenever it contains a product it contains one of the factors. The zero ideal has that property just for integral domains.

answered Dec 23 '18 at 19:22

Ethan Bolker

So the zero ideal is prime in the integers. – argiriskar Dec 23 '18 at 19:23
3

$I=0$ is a prime ideal iff $\Bbb{Z}/I=\Bbb{Z}$ is an integral domain. So yes. – Dietrich Burde Dec 23 '18 at 19:24

1 Answers1