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 $A$:

$\operatorname{Max}\:\operatorname{Spec} \left({A}\right) = \left\{{\mathfrak m \lhd A : \mathfrak m \text{ is maximal}}\right\}$

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


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

$(\operatorname{Max}\:\operatorname{Spec} \left({A}\right), \tau, \mathcal O_{\operatorname{MaxSpec}(A)})$

where:

$\tau$ is the Zariski topology on $\operatorname{MaxSpec}(A)$
$\mathcal O_{\operatorname{MaxSpec}(A)}$ is the structure sheaf of $\operatorname{MaxSpec}(A)$


Also denoted as

The maximal spectrum of $A$ can also be denoted $\operatorname{max}(A)$.


Also see