Let $(R,+,\cdot)$ be a commutative ring. It is a well-known fact that the ring of $R$-module endomorphisms of $R$, that is $(\operatorname{End}_R(R),+,\circ)$, is isomorphic to $(R,+,\cdot)$. Clearly, every module homomorphism is a homomorphism of abelian groups (and thus a $\mathbb Z$-module homomorphism), and thus we have $\operatorname{End}_R(R) \subseteq \operatorname{End}_{\mathbb Z}(R)$. In other words: The map $R \to \operatorname{End}_{\mathbb Z}(R)$ given by $r\mapsto(x\mapsto rx)$ is an injective ring homomorphism. My question is: Under which conditions on $R$ is this map also surjective?
[According to the comments, the following question can be answered by "no": Are rings $R$ where $(R,+)$ is cyclic the only examples?]