Maximal regular right ideal space of a primitive ring. II

Abstract

If R R is a ring, let X ( R ) X(R) be the set of maximal regular right ideals of R R . For each nonempty subset E E of R R , define the hull of E E to be the set { I ϵ X ( R ) | E ⊆ I } \{ I \epsilon \, X(R)|\ E \subseteq I\} and the support of E E to be the complement of the hull of E E . Topologize X ( R ) X(R) by taking the supports of right ideals of R R as a subbase. If R R is a right primitive ring, then X ( R ) X(R) is homeomorphic to an open subset of a compact space X ( R # ) X({R^\# }) of a right primitive ring R # {R^\# } , and X ( R ) X(R) is a discrete space if and only if X ( R ) X(R) is a compact Hausdorff space if and only if either R R is a finite ring or a division ring. Call a closed subset F F of X ( R ) X(R) a line if F F is the hull of I ∩ J I \cap J for some two distinct elements I I and J J in X ( R ) X(R) . If R R is a semisimple ring, then every line contains an infinite number of points if and only if either R R is a division ring or R R is a dense ring of linear transformations of a vector space of dimension two or more over an infinite division ring such that every pair of simple (right) R R -modules are isomorphic.