Lemma 7

Suppose that A admits a unit element. Let I be a maximal two-sided ideal of A. Then I is a primitive ideal.

There exists a maximal left ideal R of A containing I (A, I, p. 99, Theorem 1). Let $\pi$ be the canonical representation of A in $A/R$, which is irreducible and nonzero. Since IA $\subset R$, the kernel $I'$ of $\pi$ contains I, hence $I'= I$ and I is primitive.

Let J(A) be the set of primitive ideals of A. For every subset M of A, we denote by V(M) the set of primitive ideals of A containing M; if I is the two-sided ideal of A generated by M, one has V(M) = V(I). If M is reduced to a single element $x$, one writes $V(x)$ instead of $V(\{x\})$. The mapping $M\mapsto V(M)$ is decreasing for the inclusion relations. One has :

for every family $(M_i)_{i\in I}$ of subsets of A. On the other hand, by lemma 6,

for every two-sided ideals $I_1,I_2$ of A. Formulas (5) to (7) show that the subsets V(M) of J(A) are the closed subsets of a topology called the Jacobson topology on J(A).

Let T be a subset of J(A), and let Υ(T) be the intersection of the elements of T, so that Υ(T) is a two-sided ideal of A. Then the closure of T in J(A) is the smallest closed subset of J(A) containing T, that is to say V(Υ(T)). In particular, T is closed if and only if T = V(Υ(T)).