On the structure of exchange rings

摘要

An associative ringRwith identity is called an exchange ring ifRR;has the exchange property introduced by Crawley and Jon- sson [5]. We prove, in this paper, that ifRis an exchange ring with prime factors Artinian, thenRis strongly π-regular. IfRis an exchange ring with primitive factors Artinian andR/J(R)is homomorphically semipimitive, thenR/J(R)is strongly π-regular and idempotents lift moduloJ(R).Also, it is shown that for exchange rings, bounded index of nilpotence implies primitive factors Artinian. These are generalizations of the corresponding results in [16], [11], [8] and [2]. Examples are given showing that the generalizations are nontrivial.