It's stated here that for a commutative ring $R$, every simple module over $R$ is isomorphic as an $R$-module to a quotient ring of $R$ by a maximal ideal.
Intuitively this seems likely, since a quotient ring is a field if any only if we quotient by a maximal ideal, and a field has no nonzero proper ideals. This makes me suspect that an $R$-module $M$ is simple if and only if $M\cong R/I$ (as $R$-modules) for a maximal ideal $I$ of $R$.
Is there a proof of whether this is true or not?