Is the divisibility symbol (vertical bar) defined for rational numbers?

Question

Can I use

$k \in \mathbb{Q}, M_{k} = \{c~|~c \in \mathbb{Q} \land k|c \}$

to mean the set of all multiples of $k$? Or in other words, is the $|$ operator defined for rational numbers?

I was confused as the German version of the wikipedia entry for the vertical bar seems to limit its applicability to integers:

https://de.wikipedia.org/wiki/Senkrechter_Strich

(for those who can read it)

Edit:

I guess I confused $x|y$ to evaluate to true for rational numbers if $\frac{x}{y} \in \mathbb{Z}$. I guess @G Tony Jacobs anticipated my mistake and recommend I use $\{ nk | n \in \mathbb{Z} \}$.

Edit:

In fact, I wanted to use the vertical bar notation as building block to express the following:

$f(x,y) = \text{the smallest number $z$ so that $z = xa, a \in \mathbb{Z} \land z = xb, b \in \mathbb{Z}$}$

Maybe there is an idiomatic mathematical notation for this?

I posted somewhat of a follow-up here: Does the following hold true and how to learn how to solve this?

Answer 1

If you want the set of all integer multiples of $k$, you should write $$\{nk:n\in\Bbb{Z}\},$$ or more compactly: $$k\Bbb{Z}.$$

If you want the set of all rational multiples of $k$, that's just $\Bbb{Q}$ again anyway, as long as $k\ne 0$.

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The divisibility relation can be defined in any commutative ring $R$, by the rule: For $a,b\in R$, we say $a|b$ if there is a $c\in R$ such that $ac=b$.

The rational numbers do form a ring, but it is also a field, and we usually don't bother talking about divisibility in a field (although it is well-defined), because every non-zero element divides every element, so there's nothing to talk about. Divisibility in interesting in $\Bbb{Z}$ and other rings that aren't fields, because some integers divide each other, and some do not!

If you're using the symbol to mean "divisibility with an integer quotient".... What you've written isn't a good notation for that, because the symbol is well-defined in any ring. Here, $\Bbb{Q}$ is the only ring mentioned in your OP, so readers would be liable to interpret the symbol as divisibility in $\Bbb{Q}$. If that is what you intend, for whatever reason, then it might still be worth specifying, for those unaccustomed to talking about divisibility in fields.

Answer 2

No, divisibility is not restricted to integers. We can define $a\mid b$ in any integral domain $R$ by the property that there exists an element $c\in R$ such that $b=ac$. This can be translated to principal ideals, i.e., to contain is to divide: we have $a\mid b$ if and only if $(a)\supseteq (b)$. Of course, if $R$ is a field, this is not very useful, as all nonzero elements are units.