Quasi-morphic rings

A ring R is called left morphic if R/Ra congruent to l(a) for each a epsilon R, equivalently if there exists b epsilon R such that Ra = l(b) and l(a) = Rb. In this paper, we ask only that b and c exist such that Ra = l(b) and l(a) = Rc, and call R left quasi-morphic if this happens for every element a of R. This class of rings contains the regular rings and the left morphic rings, and it is shown that finite intersections of principal left ideals in such a ring are again principal. It is further proved that if R is quasi-morphic (left and right), then R is a Bezout ring and has the ACC on principal left ideals if and only if it is an artinian principal ideal ring.