EXCHANGE RINGS WITH GENERAL ALEPH-NOUGHT-COMPARABILITY

Abstract

In this paper, the exchange ring R with the (general) ℵ0-comparability is studied. A ring R is said to satisfy the general ℵ0-comparability, if for any idempotent elements f, g ∈ R, there exist a positive integer n and a central idempotent element e ∈ R such that f Re ≼⊕ n[gRe] and gR(1 − e) ⪯⊕ n[f R(1 − e)]. It is proved that the (general) ℵ0-comparability for exchange rings is preserved under taking factor rings, matrix rings and corners. The ℵ0-comparability condition for exchange rings R is characterized by the order structure of several partially ordered sets of ideals of R. For any exchange ring R with general ℵ0-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) There exists an infinite set of nonzero idempotent elements {f i ∣ i = 1,2, …} in I such that f 1 R ⊇⊕ f 2 R ⊇⊕ …, and n(f n R) ≼⊕ R R for all n ≥ 1; 2) For any m ≥ 1, there exist nonzero orthogonal idempotents e 1, e 2 …, e m in I such that e 1 R ⊕ e 2 R ⊕ … ⊕ e m R ⊆⊕ I R and e i R ≅ e j R for all i, j. For any exchange ring R with primitive factor rings artinian, if R satisfies the general ℵ0-comparability, then in every ideal I of R not contained in J(R), there is a central idempotent element of R.