Minimal primal ideals in rings and Banach algebras (Q1818623)

Let \(R\) be a semiprime ring. It is shown that \(\text{MinSpec}(R)\), the space of minimal primal ideals of \(R\), is compact if and only if for each principal ideal \(I\) of \(R\) there exist finitely-generated ideals \(I_1,I_2,\dots, I_n\) such that \(I^{\perp\perp}= (I_1I_2\cdots I_n)^\perp\), and that \(\text{MinSpec}(R)\) is compact and extremally disconnected if and only if the same is true for all ideals \(I\) of \(R\). These results follow from analogous ones for \(0\)-distributive, algebraic lattices. If \(R\) is a countable, semiprime ring then the set of minimal primal ideals which are prime is dense in \(\text{MinSpec}(R)\). If \(R\) is a semiprime Banach algebra in which every family of mutually orthogonal ideals is countable, then \(\text{MinSpec}(R)\) is compact and extremally disconnected, and every minimal primal ideal of \(R\) is prime.