Exchange rings with general aleph-nought-comparability

In this paper, the exchange ring R with the (general) aleph (0)-comparabili ty is studied. A ring R is said to satisfy the general aleph (0)-comparabil ity, if for any idempotent elements f,g is an element of R, there exist a p ositive integer n and a central idempotent element e is an element of R suc h that f Re less than or equal to (circle plus) n[gRe] and gR(1 - e) less t han or equal to (circle plus) n[fR(1 - e)] It is proved that the (general) aleph (0)-comparability for exchange rings is preserved under taking factor rings, matrix rings and corners. The aleph (0)-comparability condition for exchange rings R is characterized by the order structure of several partia lly ordered sets of ideals of R. For any exchange ring R with general aleph (0)-comparability and any proper ideal I of R not contained in J(R), it is proved that if I contains no nonzero central idempotents of R, then: 1) Th ere exists an infinite set of nonzero idempotent elements {f(i) / i = 1, 2, ...} in I such that f(1) R superset of or equal to (circle plus) f(2) R sup erset of or equal to (circle plus) and n(f(n) R) less than or equal to (cir cle plus) R-R for all n greater than or equal to 1; 2) For any m greater th an or equal to 1, there exist nonzero orthogonal idempotents e(1), e(2),... e(m) in I such that e(1) R circle plus e(2) R circle plus...circle plus e(m ) R subset of or equal to (circle plus) I-R and e(i) R congruent to e(j) R for all i, j. For any exchange ring R with primitive factor rings artinian, if R satisfies the general aleph (0)-comparability.