If $x + y \sim x$ in a commutative monoid, does this imply $y \sim 0$?

Question

Let $(M,0,+)$ be a commutative monoid. A congruence relation is an equivalence relation, such that $$ a \sim b, c \sim d \quad \mbox{implies} \quad a + c \sim b + d. $$ for all $a,b,c,d \in M$.

Fix some $x,y \in M$. Does $x + y \sim x$ imply $y \sim 0$? Do you know a counter-example?