Definition:Prime Spectrum of Ring

Definition
Let $A$ be a commutative ring with unity.

The prime spectrum or spectrum of $A$ is the set of prime ideals of $A$:


 * $\operatorname{Spec}(A) = \{\mathfrak p \lhd A : \mathfrak p \text{ is prime}\}$

The notation $\operatorname{Spec}(A)$ is also a shorthand for the locally ringed space
 * $(\operatorname{Spec}(A), \tau, \mathcal O_{\operatorname{Spec}(A)})$

where:
 * $\tau$ is the Zariski topology on $\operatorname{Spec}(A)$
 * $\mathcal O_{\operatorname{Spec}(A)}$ is the structure sheaf of $\operatorname{Spec}(A)$

Also see

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