\begin{definition}[Definition:Maximal Ideal of Ring]
Let $R$ be a ring.
An ideal $J$ of $R$ is '''maximal''' {{iff}}:
:$(1): \quad J \subsetneq R$
:$(2): \quad$ There is no ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.
That is, {{iff}} $J$ is a maximal element of the set of all proper ideals of $R$ ordered by inclusion.
\end{definition}