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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 $\operatorname {supp} (A) = \{ I \in X(R)|A \nsubseteq I\}$. 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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Keywords: Maximal regular right ideals, socle, reducible spaces, Hausdorff spaces, support
Article copyright: © Copyright 1972 American Mathematical Society