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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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DOI: https://doi.org/10.1090/S0002-9947-1975-0360663-6
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