If $P$ a projective $R$-module is simple, then it is a direct summand of $R$ itself (indeed, any nonzero map $R\to P$ is surjective and therefore splits) and is therefore isomorphic to a minimal ideal $I$ which is projective. $I$ is generated by an idempotent, which is central if the ring is commutative. It follows that $I$ is a direct factor of $R$ as a ring, and therefore $I$ is also injective.

If $P$ a projective $R$-module is simple, then it is a direct summand of $R$ itself (indeed, any nonzero map $R\to P$ is surjective and therefore splits) and is then isomorphic to a minimal ideal $I$ which is projective. $I$ is generated by an idempotent, which is central if the ring is commutative. It follows that $I$ is a direct factor of $R$ as a ring, and therefore $I$ is also injective.