Maximal ideal

Let $R$ be a commutative ring with $1\neq 0$. A maximal ideal $M\subsetneq R$ is a proper ideal such that there is no ideal $I$ with $M\subsetneq I\subsetneq R$.

Maximal ideals are characterized by quotients: $M$ is maximal if and only if the quotient ring $R/M$ is a field . In commutative algebra, every maximal ideal is prime .

Examples:

• In $\mathbb{Z}$, the ideal $(p)$ is maximal exactly when $p$ is a prime integer; then $\mathbb{Z}/(p)\cong \mathbb{F}_p$.
• If $k$ is a field, then in $k[x]$ the ideal $(x-a)$ is maximal for any $a\in k$.
• The ideal $(0)\subset \mathbb{Z}$ is not maximal since $(0)\subsetneq (2)\subsetneq \mathbb{Z}$.