Maximal ideal

Definition for commutative rings

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 $M$ ≤ $J$ ≤ $R$, $J$ is equal either to $M$ or to $R$.
• The quotient ring $R/M$ is a field.

Definition for non-commutative rings

For non-commutative rings, there are three notions:

• Maximal two-sided ideal: Maximal among two-sided ideals
• Maximal left ideal: Maximal among left ideals
• Maximal right ideal: Maximal among right ideals