Why is a semisimple (Wedderburn) ring von Neumann regular? In exercise 6 of chapter 5 Passman's "A Course in Ring Theory" he asks us to prove that every semisimple (Or Wedderburn as he calls it) ring is von Neumann regular. It is easy to show that a direct product of von Neumann regular rings remains von Neumann regular, so by the Artin-Wedderburn theorem I just need to show that a full matrix ring over a division ring is von Neumann regular. I believe I can do this by proving that $\operatorname{End}_DV$, where $V$ is a right module over a divison ring $D$ , is von Neumann regular, and then using the fact that $M_n(D)\cong \operatorname{End}_DD^n$. However, this way seems rather long, and I was wondering if there is a slicker way of showing that all semisimple rings are von Neumann regular?

A ring $R$ is von Neumann regular if and only if each principal left ideal is generated by an idempotent. In a semisimple ring, every left ideal is a direct summand.