I know that if two $R$-modules are isomorphic i.e. $M_1=Rm_1, M_2=Rm_2$ and $Rm_1\cong Rm_2$ then we have that $R/Ann(m_1)\cong R/Ann(m_2)$(Ann(m) is just the set of annihilators of $m$). Since $R$ is commutative this implies that $Ann(m_1)=Ann(m_2)$. Also, for any ideal $I$ of $R$ we have a cyclic $R$-module i.e. $R/I=R(1+I)$ and $Ann(1+I)=I$. Hence, the set of equivalent classes of isomorphic cyclic $R$-modules is in bijection with the ideals of $R$.
Does anyone know if that is the correct classification?