Definitions of a Factor Surrounding $0 | 0$

Question

As far as I can tell, there seems to be some controversy surrounding whether $0 | 0$. Is this partly due to different definitions of what it means for, say, $a$ to be a factor of $b$?

[Def 1]: $\forall a, b\in \mathbb{Z}$, $a | b \iff \exists c\in\mathbb{Z}\;(b = ca)$.

• This would allow $0|0$ since $0 = 0 \cdot 0$.

[Def 2]: $\forall a, b \in \mathbb{Z}$, $a | b \iff {b \over a} \in \mathbb{Z}$.

• This would not allow $0|0$ since ${0 \over 0} \not \in \mathbb{Z}.$

So which is the most common definition of a factor? I would like to know whether I can or should use $0|0$ in a slightly unrelated proof.

Answer 1

In number theory, the first of your definitions is practically universal (though $c$ should be existentially quantified, as noted by Milo Brandt in a comment).

It implies that $a\mid a$ for all $a$, that everything divides $0$, and that $a\mid b$ exactly if $\langle b\rangle\subseteq \langle a\rangle$ (as an inclusion between principal ideals of $\mathbb Z$).

Also, once you accept $a\mid 0$ for nonzero $a$, it is necessary that $0\mid 0$, or divisibility would not be a partial order.

Answer 2

Definition 2 goes into the idea that $\mathbb{Z}$ somehow admits division but is not a group, so is not probably a good idea.

Definition 1 should read:

$$\forall a,b \in\mathbb{R} a | b \iff \exists c\in\mathbb{Z}\;(b = ca), \text{with c unique}$$

Which as we have $0=c\cdot0$ as true, $c$ is not unique.