Minimal primal ideals in rings and Banach algebras Original Research Article

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 I1,I2,…,In such that Iperpendicularperpendicular=(I1I2…In)perpendicular, 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 MinSpec(R) is compact and extremally disconnected, and every minimal primal ideal of R is prime.