Maximal Ideal

definitionring-theory

Definition

For a commutative ring $R$ with $1$. An ideal $m⊴R$ is maximal if $m\ne R$, and $m\subseteq I⊴R$, then $I=m$ or $I=R$.

Examples

• In $\mathbb{Z}$, $m\ge 2$ maximal if and only if $m$ is a prime integer.