Elements of the form a/b in an integral domain - simple question

[Math Amateur]

In Alaca and Williams' (A&W) book: Introductory Algebraic Number Theory , Theorem 1.2.1 reads as follows:View attachment 3456Without any earlier definition or clarification, A&W refer to \(\displaystyle a/p\) and \(\displaystyle b/p\) (see text above) ... BUT ... how should we regard such elements? What exactly do they mean and how do we know they exist?

That is, is \(\displaystyle a/p\) simply shorthand for an element \(\displaystyle x\) where \(\displaystyle a = px\)?

Further, presumably using the notation \(\displaystyle a/p\) implies that \(\displaystyle p^{-1}\) exists ... BUT what if it does not exist?

Hope someone can clarify these apparently simple matters ...

Peter

 

[caffeinemachine]

In Alaca and Williams' (A&W) book: Introductory Algebraic Number Theory , Theorem 1.2.1 reads as follows:View attachment 3456Without any earlier definition or clarification, A&W refer to \(\displaystyle a/p\) and \(\displaystyle b/p\) (see text above) ... BUT ... how should we regard such elements? What exactly do they mean and how do we know they exist?

That is, is \(\displaystyle a/p\) simply shorthand for an element \(\displaystyle x\) where \(\displaystyle a = px\)?

Further, presumably using the notation \(\displaystyle a/p\) implies that \(\displaystyle p^{-1}\) exists ... BUT what if it does not exist?

Hope someone can clarify these apparently simple matters ...

Peter

Since $D$ is an integral domain, and $p\neq 0$, there is a unique $c\in D$ such that $pc=a$ if $p|a$.

So when we write $a/p$, I think we mean $c$.