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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
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by Kwangil Koh and Jiang Luh PDF
Trans. Amer. Math. Soc. 170 (1972), 269-277 Request permission

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.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 170 (1972), 269-277
  • MSC: Primary 16A20
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0304413-5
  • MathSciNet review: 0304413