# Spectrum of a ring

A topological space $\mathop{\rm Spec} A$ whose points are the prime ideals $\mathfrak p$ of a ring $A$ with the Zariski topology (also called the spectral topology). It is assumed that $A$ is commutative and has an identity. The elements of $A$ can be regarded as functions on $\mathop{\rm Spec} A$ by setting $a( \mathfrak p ) \equiv a$ $\mathop{\rm mod} \mathfrak p \in A / \mathfrak p$. $\mathop{\rm Spec} A$ supports a sheaf of local rings ${\mathcal O} ( \mathop{\rm Spec} A)$, called its structure sheaf. For a point $\mathfrak p \in \mathop{\rm Spec} A$, the stalk of ${\mathcal O} ( \mathop{\rm Spec} A )$ over $\mathfrak p$ is the localization $A _ {\mathfrak p}$ of $A$ at $\mathfrak p$.

To any identity-preserving ring homomorphism $\phi : A \rightarrow A ^ \prime$ there corresponds a continuous mapping $\phi ^ {*} : \mathop{\rm Spec} A ^ \prime \rightarrow \mathop{\rm Spec} A$. If $N$ is the nil radical of $A$, then the natural mapping $\mathop{\rm Spec} A /N \rightarrow \mathop{\rm Spec} A$ is a homeomorphism of topological spaces.

For a non-nilpotent element $f \in A$, let $D( f )= ( \mathop{\rm Spec} A ) \setminus V( f )$, where $V( f )= \{ {\mathfrak p \in \mathop{\rm Spec} A } : {f \in \mathfrak p } \}$. Then the ringed spaces $D( f )$ and $\mathop{\rm Spec} A _ {(} f)$, where $A _ {(} f)$ is the localization of $A$ with respect to $f$, are isomorphic. The sets $D( f )$ are called the principal open sets. They form a basis for the topology on $\mathop{\rm Spec} A$. A point $\mathfrak p \in \mathop{\rm Spec} A$ is closed if and only if $\mathfrak p$ is a maximal ideal of $A$. By assigning to $\mathfrak p$ its closure $\overline{ {\mathfrak p }}\;$ in $\mathop{\rm Spec} A$, one obtains a one-to-one correspondence between the points of $\mathop{\rm Spec} A$ and the set of closed irreducible subsets of $\mathop{\rm Spec} A$. $\mathop{\rm Spec} A$ is quasi-compact, but usually not Hausdorff. The dimension of $\mathop{\rm Spec} A$ is defined as the largest $n$ for which there is a sequence of distinct closed irreducible sets $Z _ {0} \subset \dots \subset Z _ {n} \subset \mathop{\rm Spec} A$.

Many properties of $A$ can be described in terms of $\mathop{\rm Spec} A$. For example, $A/N$ is Noetherian if and only if $\mathop{\rm Spec} A$ has the descending chain condition for closed sets; $\mathop{\rm Spec} A$ is an irreducible space if and only if $A/N$ is an integral domain; the dimension of $\mathop{\rm Spec} A$ coincides with the Krull dimension of $A$, etc.

Sometimes one considers the maximal spectrum $\mathop{\rm Specm} A$, which is the subspace of $\mathop{\rm Spec} A$ consisting of the closed points. For a graded ring $A$ one also considers the projective spectrum $\mathop{\rm Proj} A$. If $A= \sum _ {n=} 0 ^ \infty A _ {n}$, then the points of $\mathop{\rm Proj} A$ are the homogeneous prime ideals $\mathfrak p$ of $A$ such that $\mathfrak p \Nps \sum _ {n=} 1 ^ \infty A _ {n}$.

How to Cite This Entry:
Spectrum of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectrum_of_a_ring&oldid=48769
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article