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