LEFT QUASI-MORPHIC RINGS

Abstract

A ring R is called left quasi-morphic if, for each a ∈ R, there exist b and c in R such that Ra = l (b) and l (a) = Rc (where l (x) is the left annihilator). Every (von Neumann) regular ring is left quasi-morphic, as is every left morphic ring (b = c above). The main theorem of this paper is that, in a left quasi-morphic ring, finite intersections and finite sums of principal left ideals are again principal. This leads to structure theorems when mild finiteness conditions are imposed. In an earlier paper, the first two authors showed that left and right quasi-morphic rings have both these properties (on both sides), and used this to give a new characterization of the artinian principal ideal rings: They are just the left and right quasi-morphic rings with ACC on principal annihilators r (a), a ∈ R. Some extensions of this result are presented here.