This CSL ("converse of Schur's Lemma") condition on a ring is actually a topic of interest in ring theory lately.  This basically means that there is not yet any simple answer to the question.  But there is some interesting progress toward partial answers.  For instance, a [recent paper][1] by G. Marks and M. Schmidmeier shows that the converse of Schur's Lemma holds *in the category of right R-modules of finite length* if and only if all extensions of simple right R-modules are split.  In particular, this holds over any commutative ring.  (The same paper cites a number of other sources if you are interested in further exploring the topic.)


  [1]: http://arxiv.org/PS_cache/arxiv/pdf/0903/0903.2490v2.pdf