Definition:Prime Spectrum of Ring

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Definition

Let $A$ be a commutative ring with unity.


The prime spectrum of $A$ is the set of prime ideals $\mathfrak p$ of $A$:

$\Spec A = \set {\mathfrak p \lhd A: \mathfrak p \text{ is prime} }$

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


Also defined as

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

$\struct {\Spec A, \tau, \OO_{\Spec A} }$

where:

$\tau$ is the Zariski topology on $\Spec A$
$\OO_{\Spec A}$ is the structure sheaf of $\Spec A$


Also known as

The prime spectrum of a commutative ring with unity is also referred to just as its spectrum.


Also see


Sources