Maximal ideal

Symbol-free definition
An ideal in a commutative unital ring (or more generally, in any commutative ring) is termed maximal if it is proper (not the whole ring) and it satisfies the following equivalent conditions:


 * There is no ideal of the ring properly in between this ideal and the whole ring
 * The quotient of the ring by this ideal is a field
 * The natural action of $$R$$ on $$R/M$$ makes $$R/M$$ into a simple $$R$$-module.

Definition with symbols
An ideal $$M$$ in a commutative ring $$R$$ is termed maximal if it satisfies the following equivalent conditions:


 * For any ideal $$J$$ such that $$M$$ &le; $$J$$ &le; $$R$$, $$J$$ is equal either to $$M$$ or to $$R$$.
 * The quotient ring $$R/M$$ is a field.

Weaker properties

 * Stronger than::Prime ideal:
 * Stronger than::Radical ideal
 * Stronger than::Intersection of maximal ideals