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 $A$:


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

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

Zariski Topology
The spectrum $\operatorname{Spec}(A)$ is rarely considered as a set: it is understood to be a topological space with the Zariski topology.

Definition 1
Note that Principal Open Subsets form Basis of Zariski Topology on Prime Spectrum.

We define the structure sheaf of $\operatorname{Spec}(A)$ to be the sheaf induced by a sheaf on this basis defined as follows:
 * For $f\in A$, $\mathcal O(X(f))$ is the localization of $A$ at $f$
 * For $f, g \in A$ with $X(f)\supset X(g)$, the restriction is the unique homomorphism of $A$-algebras $A_f \to A_g$.

Also see

 * Definition:Maximal Spectrum of Ring
 * Definition:Affine Scheme