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Transactions of the American Mathematical Society

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

 

 

Maximal regular right ideal space of a primitive ring


Authors: Kwangil Koh and Jiang Luh
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
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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)\vert A \nsubseteq I\} $. We give a topology to $ X(R)$ by taking $ \{ \operatorname{supp} (A)\vert 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

DOI: https://doi.org/10.1090/S0002-9947-1972-0304413-5
Keywords: Maximal regular right ideals, socle, reducible spaces, Hausdorff spaces, support
Article copyright: © Copyright 1972 American Mathematical Society