Maximal regular right ideal space of a primitive ring. II

Authors:
Kwangil Koh and Hang Luh

Journal:
Trans. Amer. Math. Soc. **180** (1973), 127-141

MSC:
Primary 16A20

DOI:
https://doi.org/10.1090/S0002-9947-1973-0338049-8

MathSciNet review:
0338049

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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.

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Keywords:
Maximal regular right ideals,
socle,
irreducible spaces,
compact Hausdorff spaces,
lines,
hyperplanes,
support

Article copyright:
© Copyright 1973
American Mathematical Society