Let $x,y$ be two generators of a cyclic module, $M=Rx=Ry$, then $ann(x)=ann(y)$.
I think that it has to be associated to the fact that if $I,J$ are ideals of a commutative ring $R$ such that exists a module isomorphism $\frac{R}{I}\cong\frac{R}{J}$ then $I=J$. But I am not sure of that, could you tell me if I am right and in that case how to apply that statement to what I want to prove?
Thank you so much for your help.
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Towards your main question: The assumption $x \in Ry = M$ should allow you to conclude $\mathrm{ann}(y) \subseteq \mathrm{ann}(x)$. Do you see where to go from there?