notation - Is modular arithmetic defined for all rational numbers (with denominators coprime to modulus)?

In the expression $\frac{1}{b}\pmod m$, where $(b,m)=1$, is $\frac{1}{b}$:

a) a rational number (and so rational numbers are defined in modulo arithmetic using multiplicative inverses)?

b) just created notation for $($the inverse of $b)\pmod m$ that looks like division just to confuse us (and is used because of similarities between division and division $\pmod m$)? (same for $b^{−1}$)

Is it a) or b)?

Here it says it is a).

Bill Dubuque from M.SE seemingly claims it is b). So does a comment here, also this blog.

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Edit: now that I thought about it, either

1) the notation $\frac{a}{b}$

2) or the definition $a\equiv b\pmod {m}\iff m\mid a-b$

is misleading. 1) seems a lot more likely, since I'm not sure how the $\bmod$ function could be defined otherwise.