Definition 4

Let $A$ be a ring. The set $X$ of prime ideals of $A$, with the topology whose closed sets are the sets $V(M)$, where $M$ runs through $\mathfrak{P}(A)$, is called the prime spectrum of $A$ and denoted by $\mathrm{Spec}(A)$. The topology thus defined is called the spectral or Zariski topology on $X$.

Clearly the relation $\mathrm{Spec}(A) = \varnothing$ is equivalent to $A = \{0\}$.

Let $X$ be the prime spectrum of a ring $A$; for all $f \in A$, let us denote by $X_f$ the set of prime ideals of $A$ not containing $f$; then $X_f = X - V(f)$ and $X_f$ is therefore an open set. By (2), every closed subset of $X$ is an intersection of closed sets of the form $V(f)$ and hence the $X$, form a base of the spectral topology on $X$. Moreover, it follows immediately from the definitions that

and more generally $X_{f^{-1}} = X$ for every invertible element $f$ of $A$;

For every subset $Y$ of $X$, let us denote by $\mathfrak{J}(Y)$ the intersection of the prime ideals of $A$ which belong to $Y$. Clearly $\mathfrak{J}(Y)$ is an ideal of $A$ and the mapping $Y \mapsto \mathfrak{J}(Y)$ is decreasing with respect to inclusion in $X$ and $A$. Clearly the relations

hold for every family $(Y_\lambda)_{\lambda \in L}$ of subsets of $X$. Moreover: