Definition:Maximal Ideal of Ring

Definition
Let $$R$$ be a ring.

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


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

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.