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In this paper a right order      R R       in an infinite dimensional full linear ring is characterized as a ring satisfying: (1)      R R       is meet-irreducible (with zero right singular ideal) and contains uniform right ideals; (2) the closed right ideals of      R R       are right annihilator ideals, and each such right ideal is essentially finitely generated; (3)      R R       possesses a reducing pair (i.e. a pair      ( β  1 , β  2 ) ({\beta _1},{\beta _2})       of elements for which      β  1 R , β  2 R {\beta _1}R,{\beta _2}R       and      β  1 r + β  2 r \beta _1^r + \beta _2^r       are large right ideals of      R R      ); (4) for each      a ∈  R a \in R       with      a l = 0 , a R {a^l} = 0,aR       contains a regular element of      R R      . A second characterization of      R R       is also given. This is in terms of the right annihilator ideals of      R R       which have the same (uniform) dimension as      R R {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.

Right orders in full linear rings