Over a commutative ring $R$, any simple module has the form $R/\mathfrak m$ for a maximal ideal $\mathfrak m$.  Hence we may localize at $\mathfrak m$ and assume $R$ is local.  If $R/\mathfrak m$ is projective, then it is a summand of $R$ (by considering the natural surjection $R\rightarrow R/\mathfrak m$).  Now $R = I\oplus R/\mathfrak m$ for some (necessarily finitely generated) ideal $I$ and $I$ is such that $I = \mathfrak m I$.  By Nakayamma's lemma, $I = 0$ and $R$, hence $R = R/\mathfrak m$ and $R$ is a field.