A module which is not singular

Question

Suppose that M is a projective R-module and that it's simple, isomorphic to $R/I$ where $I$ is a maximal left ideal of $R$ such that $I$ is not a direct summand of $R$. How to get to a contradiction?

Answer 1

If a module $P$ is projective then for a surjective module homomorphism $f : R \to P$ there exists a module homomorphism $h : P \to R$ such that $fh = {\rm id}$. So $P$ is a direct summand of $R$. More explicitly, $R = {\rm Im} (h) \oplus I$. The contradiction.