# Definition:Maximal Spectrum of Ring

## Definition

Let $A$ be a commutative ring with unity.

The **maximal spectrum** of $A$ is the set of maximal ideals of $A$:

- $\operatorname{Max} \: \Spec A = \set {\mathfrak m \lhd A : \mathfrak m \text { is maximal} }$

where $I \lhd A$ indicates that $I$ is an ideal of $A$.

The notation $\operatorname {Max} \: \Spec A$ is also a shorthand for the locally ringed space

- $\struct {\operatorname {Max} \: \Spec A, \tau, \mathcal O_{\map {\operatorname {Max Spec} } A} }$

where:

- $\tau$ is the Zariski topology on $\map {\operatorname {Max Spec} } A$
- $\mathcal O_{\map {\operatorname {Max Spec} } A}$ is the structure sheaf of $\map {\operatorname {Max Spec} } A$

## Also denoted as

The **maximal spectrum** of $A$ can also be denoted $\map {\max} A$.