Maximal regular right ideal space of a primitive ring

Abstract:

If $R$ is a ring, let $X(R)$ be the set of maximal regular right ideals of $R$ and $\mathfrak {L}(R)$ be the lattice of right ideals. For each $A \in \mathfrak {L}(R)$, define $\operatorname {supp} (A) = \{ I \in X(R)|A \nsubseteq I\}$. We give a topology to $X(R)$ by taking $\{ \operatorname {supp} (A)|A \in \mathfrak {L}(R)\}$ as a subbase. Let $R$ 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. $X(R)$ is a Hausdorff space if and only if either $R$ is a division ring or $R$ 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.