Definition:Maximal Spectrum of Ring
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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
- $\struct {\operatorname {Max} \: \Spec A, \tau, \OO_{\map {\operatorname {Max Spec} } A} }$
where:
- $\tau$ is the Zariski topology on $\map {\operatorname {Max Spec} } A$
- $\OO_{\map {\operatorname {Max Spec} } A}$ is the structure sheaf of $\map {\operatorname {Max Spec} } A$
Also denoted as
The maximal spectrum of $A$ can also be denoted $\map \max A$.