REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: Two elements a,b in a ring R form a right coprime pair when they satisfy the right Bezout identity, i.e., aR + bR = R. Coprime pairs have shown to be quite useful in the study of exchange rings and cotorsion rings. In this talk, we define right strongly exchange rings in terms of descending chains of right coprime pairs and show that most classes of rings that have shown to have well behaved decompositions into direct summands satisfy this condition. Namely, we show that the class of right strongly exchange rings includes left self-injective, left pure-injective, left cotorsion, left continuous, left perfect or local rings and we prove that this strongly exchange property is responsible for this good behavior. This allows us to give a unified study of the behavior of these classes of rings.