Let $R$ be a commutative ring with unit and let $M$ be an $R$-module. Then the Nagata idealization/trivialization construction $R(+)M$ is well known. Suppose $J$ is an ideal of $R(+)M$. Then it is known that $J=I(+)C$, where $I$ is an ideal of $R$ defined by the first coordinate of $J$ and $C$ is a submodule of $M$ defined by the second coordinate of $J$.
QUESTION: I am unable to show that $I(+)C$ is contained in $J$.
This should be easy but i am missing something simple. Any help is appreciated. Thanks.