Definition:Maximal Ideal of Ring

Definition
Let $R$ be a ring.

An ideal $J$ of $R$ is maximal iff:


 * 1) $J \subsetneq R$;
 * 2) There is no ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.

That is, if $J$ is a maximal element of the set of all proper ideals of $R$ ordered by $\subseteq$.

Variant Definitions
Some sources insist that $R$ be a commutative ring with unity for this definition to hold.