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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 $x=-a$?

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Being cautious, on can only conclude from $[x]=0$ that $x=a-b$ for some $a,b\in M$. Consider the case $G=\mathbb Z$ and $M=\mathbb N_0\setminus\{1\}$ (both with addition). Then $\langle M\rangle=G$ and hence $[1]=0$, but neither $1\in M$ not $-1\in M$.