In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring ${\displaystyle R}$ is the set of all prime ideals of ${\displaystyle R}$, and is usually denoted by ${\displaystyle \operatorname {Spec} {R}}$; in algebraic geometry it is simultaneously a topological space equipped with a sheaf of rings.

For any ideal ${\displaystyle I}$ of ${\displaystyle R}$, define ${\displaystyle V_{I}}$ to be the set of prime ideals containing ${\displaystyle I}$. We can put a topology on ${\displaystyle \operatorname {Spec} (R)}$ by defining the collection of closed sets to be

This topology is called the Zariski topology.

A basis for the Zariski topology can be constructed as follows: For ${\displaystyle f\in R}$, define ${\displaystyle D_{f}}$ to be the set of prime ideals of ${\displaystyle R}$ not containing ${\displaystyle f}$. Then each ${\displaystyle D_{f}}$ is an open subset of ${\displaystyle \operatorname {Spec} (R)}$, and ${\displaystyle {\big \{}D_{f}:f\in R{\big \}}}$ is a basis for the Zariski topology.

${\displaystyle \operatorname {Spec} (R)}$ is a compact space, but almost never Hausdorff: In fact, the maximal ideals in ${\displaystyle R}$ are precisely the closed points in this topology. By the same reasoning, ${\displaystyle \operatorname {Spec} (R)}$ is not, in general, a T₁ space. However, ${\displaystyle \operatorname {Spec} (R)}$ is always a Kolmogorov space (satisfies the T₀ axiom); it is also a spectral space.

Given the space ${\displaystyle X=\operatorname {Spec} (R)}$ with the Zariski topology, the structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$ is defined on the distinguished open subsets ${\displaystyle D_{f}}$ by setting ${\displaystyle \Gamma (D_{f},{\mathcal {O}}_{X})=R_{f},}$ the localization of ${\displaystyle R}$ by the powers of ${\displaystyle f}$. It can be shown that this defines a B-sheaf and therefore that it defines a sheaf. In more detail, the distinguished open subsets are a basis of the Zariski topology, so for an arbitrary open set ${\displaystyle U}$, written as the union of ${\textstyle U=\bigcup _{i\in I}D_{f_{i}}}$, we set ${\textstyle \Gamma (U,{\mathcal {O}}_{X})=\varprojlim _{i\in I}R_{f_{i}},}$ where ${\displaystyle \varprojlim }$ denotes the inverse limit with respect to the natural ring homomorphisms ${\displaystyle R_{f}\to R_{fg}.}$ One may check that this presheaf is a sheaf, so ${\displaystyle \operatorname {Spec} (R)}$ is a ringed space. Any ringed space isomorphic to one of this form is called an affine scheme. General schemes are obtained by gluing affine schemes together.

Similarly, for a module ${\displaystyle M}$ over the ring ${\displaystyle R}$, we may define a sheaf ${\displaystyle {\widetilde {M}}}$ on ${\displaystyle \operatorname {Spec} (R)}$. On the distinguished open subsets set ${\displaystyle \Gamma (D_{f},{\widetilde {M}})=M_{f},}$ using the localization of a module. As above, this construction extends to a presheaf on all open subsets of ${\displaystyle \operatorname {Spec} (R)}$ and satisfies the gluing axiom. A sheaf of this form is called a quasicoherent sheaf.

If ${\displaystyle {\mathfrak {p}}}$ is a point in ${\displaystyle \operatorname {Spec} (R)}$, that is, a prime ideal, then the stalk of the structure sheaf at ${\displaystyle {\mathfrak {p}}}$ equals the localization of ${\displaystyle R}$ at the ideal ${\displaystyle {\mathfrak {p}}}$, which is generally denoted ${\displaystyle R_{\mathfrak {p}}}$, and this is a local ring. Consequently, ${\displaystyle \operatorname {Spec} (R)}$ is a locally ringed space.