Maximal regular right ideal space of a primitive ring. II

If 7? is a ring, let X(R) be the set of maximal regular right ideals of 7? and £(7?) be the lattice of right ideals. For each A £ £(7?), define supp(/l) = J7 £ X(R)\A (7 /}. We give a topology to X(7?) by taking [supp (A) | A £ £(R)\ as a subbase. Let 7? be a right primitive ring. Then X(R) is the union of two proper closed sets if and only if R is isomorphic to a dense ring with nonzero socle of linear transformations of a vector space of dimension two or more over a finite field. X(7?) is a Hausdorff space if and only if either 7? is a division ring or 7? modulo its socle is a radical ring and R is isomorphic to a dense ring of linear transformations of a vector space of dimension two or more over a finite field. Introduction. For a ring R, define X(R) to be the set of maximal regular right ideals of R. Then X(R) is a nonempty set if and only if R is not a radical ring. If A is a right ideal of a ring R, define the support of A to be the set of maximal regular right ideals of R which do not contain A. We topologize X(R) by defining that a subset is open if and only if it is an arbitrary union of finite intersections of the supports of right ideals in R; that is, the supports of the right ideals form a subbasis for this topology. We will call X(R) together with this topology the maximal regular right ideal space of the ring 7?. Recall that a topological space is irreducible (refer to [3, p. 13]) if it is not the union of two proper closed subsets, and it is reducible if it is not irreducible. Our main results in this paper are as follows: Let R be a (right) primitive ring. Then X(7?) is reducible if and only if R is isomorphic to a dense ring with nonzero socle of linear transformations of a vector space of dimension two or more over a finite field. X(R) is a Hausdorff space if and only if either R is a division ring or 7? is isomorphic to a dense ring of linear transformations of a vector space over a finite field such that R modulo its socle is a radical ring. If R has 1, then X(R) is a Hausdorff space if and only if either R is a division ring or a finite ring. 1. Preliminaries. 1.1 Definition. If A is a right ideal of a ring R, the support of A is the Presented to the Society, May 10, 1971; received by the editors May 13, 1971. AMS 1970 subject classifications. Primary 16A20, 16A42; Secondary 16A48.