Given an abelian group $G$, one can form the endomorphism ring $\mbox{End}(x)$ \mbox{End}(G)$ by letting $\alpha+\beta=\alpha(x)+\beta(x)$, and $\alpha\beta=\alpha(\beta(x))$, where $\alpha$ and $\beta$ are endomorphisms. Clearly, composition distributes over addition, and addition is commutative, so $\mbox{End}(x)$ \mbox{End}(G)$ is a ring. My question is: when is $\mbox{End}(x)$ \mbox{End}(G)$ commutative? Are there a nice set of criterioncriteria, or, if there is no such nice set of criterioncriteria, is there a nice class of abelian groups with commutative endomorphism rings.

Given an abelian group $G$, one can form the endomorphism ring $\mbox{End}(x)$ by letting $\alpha+\beta=\alpha(x)+\beta(x)$, and $\alpha\beta=\alpha(\beta(x))$, where $\alpha$ and $\beta$ are endomorphisms. Clearly, composition distributes over addition, and addition is commutative, so $\mbox{End}(x)$ is a ring. My question is: when is $\mbox{End}(x)$ commutative? Are there a nice set of criterion, or, if there is no such nice set of criterion, is there a nice class of abelian groups with commutative endomorphism rings.