Let $A$ be a torsion-free Abelian group, and let $\mbox{End}(A)$ be its endomorphism ring. Denote by $R\doteq\mathbf{Z}(\mbox{End}(A))$ the center of $\mbox{End}(A)$. What would be necessary and/or sufficient conditions on $A$ for $R$ to be a PID?

In other words, what do I need to impose on $A$ in order to make it a PID module? If $A$ is a $P$-module with $P$ a PID, then $P\subset R\subset Q(P)$ (the quotient field, am I right?), and so $R$ is a PID. Thank you.