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Transactions of the American Mathematical Society

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



Right orders in full linear rings

Author: K. C. O’Meara
Journal: Trans. Amer. Math. Soc. 203 (1975), 299-318
MSC: Primary 16A18
MathSciNet review: 0360663
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Abstract: In this paper a right order $R$ in an infinite dimensional full linear ring is characterized as a ring satisfying: (1) $R$ is meet-irreducible (with zero right singular ideal) and contains uniform right ideals; (2) the closed right ideals of $R$ are right annihilator ideals, and each such right ideal is essentially finitely generated; (3) $R$ possesses a reducing pair (i.e. a pair $({\beta _1},{\beta _2})$ of elements for which ${\beta _1}R,{\beta _2}R$ and $\beta _1^r + \beta _2^r$ are large right ideals of $R$); (4) for each $a \in R$ with ${a^l} = 0,aR$ contains a regular element of $R$. A second characterization of $R$ is also given. This is in terms of the right annihilator ideals of $R$ which have the same (uniform) dimension as ${R_R}$. The problem of characterizing right orders in (infinite dimensional) full linear rings was posed by Carl Faith. The Goldie theorems settled the finite dimensional case.

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Keywords: Full linear ring, right order, right quasi-order, irreducible ring, uniform right ideal, uniform dimension, closed right ideal, right annihilator ideal, essentially finitely generated, reducing pair, flat epimorphic extension, idealizer
Article copyright: © Copyright 1975 American Mathematical Society