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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Maximal regular right ideal space of a primitive ring. II
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by Kwangil Koh and Hang Luh PDF
Trans. Amer. Math. Soc. 180 (1973), 127-141 Request permission

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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • 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