Direct sum of $R$ - Modules

Question

This maybe trivial but if a commutative ring $R$ is isomorphic to the direct sum of one of its ideals and some $R$ - module, is this $R$ - module isomorphic to an ideal of $R$?

Answer 1

If there is an $R$-module isomorphism $\phi:M\oplus N\to R$, with $M$ and $N$ $R$-modules, then $\phi(M\oplus 0)$ is an ideal of $R$, isomorphic to $M$ as an $R$-module.