A definition of a maximal ideal in a ring $R$ is as follows:
$I$ is a maximal ideal if for any ideal $J$ with $I \subseteq J$, either $J = I$ or $J = R$.
Let's say $I \subseteq J = R$, thus by this definition, $I$ is a maximal ideal. However $J$ is also an ideal, with size larger than that of $I$ (since $J=R$). Thus using this definition, both $J$ and $I$ are ideals, where $I$ is maximal, while being a subset of $J$. How can this be?