Maximal regular right ideal space of a primitive ring. II

Abstract:

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