Let A be an algebra. In this chapter we will study the space 〈 PA / ΠA / ΞA 〉 of 〈 prime / primitive / maximal modular 〉 ideals of A as a topological space under the hull–kernel or Jacobson topology. For certain classes of algebras A, e.g., completely regular algebras (Section 7.2) and strongly harmonic algebras (Section 7.4), we will show that the subdirect product representation relative to ΞA (introduced in Definition 1.3.3 and Section 4.6) yields significant information about A. Section 7.3 deals with more detailed questions in ideal theory revolving around primary ideals. We also consider central and weakly central algebras and show that they are completely regular under fairly weak additional hypotheses.

The Hull-Kernel Topology

In Section 3.2 we introduced the hull–kernel topology on the Gelfand space ΓA of a commutative Banach algebra A. It is comparatively little used except in the case of completely regular commutative spectral algebras where it is Hausdorff and coincides with the Gelfand topology. In the commutative case, Proposition 3.1.3 shows that the Gelfand space of A can be identified with the set ΞA of maximal modular ideals of A, and Theorem 4.1.9 shows that the latter set coincides with the set ΠA of primitive ideals.

In a noncommutative algebra A, the set PA of prime ideals and its subsets, ΠA and ΞA, can each be given the hull–kernel topology. Again, this topology seems to be of comparatively little use unless further restrictions are placed on the algebra.