Exchange general rings

摘要

The exchange rings without unity, first introduced by Ara, are further investigated. Some new characterizations and properties of exchange general rings are given. For example, a general ring I is exchange if and only if for any left ideal L of I and a = a~2 ∈ I/L, there exists w ∈ r. ureg(I) such that w = a;E(R,I) (the ideal extension of a ring R by its ideal I) is an exchange ring if and only if R and I are both exchange. Furthermore, it is presented that if I is a two-sided ideal of a unital ring R and I is an exchange general ring, then every central element of I is a clean element in I.