Right orders in full linear rings

O’Meara, K. C.

doi:10.1090/S0002-9947-1975-0360663-6

Right orders in full linear rings

Author: K. C. O’Meara
Journal: Trans. Amer. Math. Soc. 203 (1975), 299-318
MSC: Primary 16A18
DOI: https://doi.org/10.1090/S0002-9947-1975-0360663-6
MathSciNet review: 0360663
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Abstract: In this paper a right order in an infinite dimensional full linear ring is characterized as a ring satisfying:

(1) is meet-irreducible (with zero right singular ideal) and contains uniform right ideals;

(2) the closed right ideals of are right annihilator ideals, and each such right ideal is essentially finitely generated;

(3) 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 );

(4) for each $a \in R$ with ${a^l} = 0,aR$ contains a regular element of .

A second characterization of is also given. This is in terms of the right annihilator ideals of 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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