Definition:Prime Spectrum of Ring

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

• Definition:Spectrum of Ring Functor
• Definition:Maximal Spectrum of Ring
• Prime Spectrum of Ring is Locally Ringed Space
• Definition:Affine Scheme