In this answer an equivalence relation on the reals is described as
$$ x \sim y \iff x - y\in \mathbb{Q} \qquad \forall x, y\in \Bbb R$$
Denoting an equivalence class as $[x] := \{y\in\Bbb R \mid y\sim x\}$, the set of equivalence classes $T:= \{[x]\mid x\in \Bbb R\}$ forms a group with an addition defined on the representatives as $[x] + [y] := [x+y]$ with $\Bbb Q$ as the neutral element and $-[x] = [-x]$ the inverse.
Question(s): Does this structure have a name and some literature? For example, would it be possible to add a non-trivial multiplication to get to a ring or field? Is it helpful (in some non-trivial interpretation of "helpful" ;-) in some field of math?
Related: this question is about the equivalence relation defined by the quotient being rational.
It's called the quotient group of the group $(\Bbb R,+)$ by the subgroup $\Bbb Q$. Usually it's indicated by $\Bbb R/\Bbb Q$.