Minimal primal ideals in rings and Banach algebras

Let R be a semiprime ring. It is shown that 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,..., I-n such that I-perpendicular to perpendicular to = (I1I2...I-n)(perpendicular to), and that 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 MinSpec(R). If R is a semiprime Banach algebra in which every family of mutually orthogonal ideals is countable, then MiniSpec(R) is compact and extremally disconnected, and every minimal primal ideal of R is prime. (C) 1999 Elsevier Science B.V. All rights reserved.