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? Thank you.

**Edit** Following user89334's comment, indeed, this is true if $A$ is a finitely generated group. The core of the question is the infinitely generated case (no restrictions on the cardinality of the generating set).