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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)\vert\ 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^\char93 })$ of a right primitive ring $ {R^\char93 }$, 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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0338049-8
Keywords: Maximal regular right ideals, socle, irreducible spaces, compact Hausdorff spaces, lines, hyperplanes, support
Article copyright: © Copyright 1973 American Mathematical Society

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