Right orders in full linear rings

O’Meara, K. C.

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

Right orders in full linear rings
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by K. C. O’Meara

Trans. Amer. Math. Soc. 203 (1975), 299-318

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

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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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