Suppose $(G,+,0)$ is an abelian group and $(M,+,0)$ is an abelian submonoid, so $M\subseteq G$, the operation of the monoid coincides with the restricted operation of the group, the units coincide and $a,b\in M\Rightarrow a+b\in M$.
Let $\langle M\rangle$ denote the subgroup of $G$ generated by $M$.
When does $[x]=0$ hold in the factor group $G/\langle M\rangle$? Is this the case iff there exists a representative $x\in G$ of $[x]$ and an element $a\in M$ with $x=a$ or $x=-a$?