 Solution

For $R>0$, the chain has one communicating class exactly when every $r_j>0$. Sufficiency follows from the positive direct transition rate between each distinct pair of states. For necessity, if $r_j=0$ and there is another state, no other state can jump to $j$, either directly or as the final step of a path, so $j$ cannot communicate with them. The one-state case also satisfies the condition because $R=r_1>0$. A single class containing the entire state space is automatically closed.