Take a commutative ring with unity (a.k.a. ring) $R$ and let $I$ be an ideal in $R$. Are the following true??
a) If $J$ is an ideal in $I$ (observed as a ring) , $J$ is not ideal in $R$(if this is true, please give me example)
b) If $J$ is a maximal ideal in $I$, then $J$ is ideal in $R$
c) If $J$ is ideal in $I$, and $I$ is maximal ideal in $R$, then $J$ is ideal in $R$