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.

I would usually take the first definition to be better, but whether or not you can use it depends on what definition is being used in the rest of the proof (although the condition should be written $\forall a,b\in \mathbb Z,\exists c\in \mathbb Z,,a|b,\Longleftrightarrow,b=ca$. Your first definition is currently not right) — Milo Brandt, Nov 08 '15 at 14:10
If $a \neq 0$ Definition 1 implies definition 2. Thus in a way you can say definition 1 is a more general definition. — advocateofnone, Nov 08 '15 at 14:13
Does this answer your question? Why would some elementary number theory notes exclude 0|0? — Bill Dubuque, Mar 14 '23 at 09:39

hmakholm left over Monica · Accepted Answer · 2015-11-08T14:20:02.497

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.

score 1 · Answer 2 · answered Nov 08 '15 at 14:24

1

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.

answered Nov 08 '15 at 14:24

JMP

22,637

Do you mean "Definition 2 should read"? Your added uniqueness requirement would prevent $0$ from dividing itself. – hmakholm left over Monica Nov 08 '15 at 14:37
i was going for the assumption that division is well-defined over $\mathbb{Z}$ – JMP Nov 08 '15 at 14:38
Yes, but that would still be for definition 2, right? Definition 1 is not based on division at all. – hmakholm left over Monica Nov 08 '15 at 14:47

Definitions of a Factor Surrounding $0 | 0$

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