Suppose we want to find $42/12$ as a quotient followed by a remainder. One approach is to write $42/12$ as a mixed number via the usual method: $$\frac{42}{12} = \frac{36+6}{12} = \frac{36}{12}+\frac{6}{12} = 3 + \frac{6}{12}$$ So the answer is $3$ with remainder $6$. However, there's something a bit dissatisfying about this. Note that to write the RHS as a normal-form mixed number, we should simplify our fraction, obtaining $$3+ \frac{1}{2}.$$ Arguably this is slightly disconcerting: to get the correct final answer of remainder $6$ and avoid the incorrect final answer of remainder $1$, we have to avoid simplifying. That's perhaps a bit strange. More fundamentally, my issue here is that we can't actually extract the remainder from the rational number we started with. In particular, if someone tells you that $x \in \mathbb{Q}$ is the rational number $42/12$ and asks you to find the remainder, there's no sensible answer to their question - the remainder could be $1$, or any non-zero multiple of $1$.
For this reason, I'm interested in the number system that's like $\mathbb{Q},$ but without the cancellation law $ax/bx = a/b$. Lets call it $\tilde{\mathbb{Q}}$. The idea behind $\tilde{\mathbb{Q}}$ is that it's a quotient of the set of all formal $\mathbb{Z}$-linear combinations of elements of the form $$\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$$ So an example element would be $$\frac{5}{3} + \frac{17}{5},$$ by which we really mean the linear combination $$5\frac{1}{3} + 17\frac{1}{5}.$$ I'm pretty sure the above set can be identified with the Dirichlet series with finitely many terms. We then take a quotient of this ring by assuming that if $a$ and $b$ are integers satisfying $a \mid b$, then letting $c$ denote the integer $b/a$, we have $\frac{b}{a} = c\frac{1}{1}.$ I'm not sure how to translate this into Dirichlet language. Anyhoo, in the resulting set $\tilde{\mathbb{Q}}$ we've just constructed, which should be a commutative ring if I'm not mistaken, we have $$\frac{42}{12} = 3 + \frac{6}{12} \neq 3 + \frac{1}{2},$$ and so we can extract the remainder of $6$ from the quotient $42/12$, as desired.
I'd be curious to learn more, so:
Question. Does this number system have a name and/or where can I learn more about it?