QB-rings And Modules Which Have Exchange Property

Replacing invertibility with quasi-invertibility in Bass first stable range condition we study a new class of rings,the QB-rings.In the QB-rings,the quasiinvertibility and the relation "≤" on the set R~r of Von Neumann regular elements in a ring R play important roles.we will give some characterization of the quasi-invertibility and the relation "≤" in R~r,and describe the relation between Von Neumann regular elements and quasi-invertible elements in R.We show that if u is a Von Neumann regular element,then there exist an element v in R such that u = uvu,v = vuv,and(1-uv)⊥(1 -vu),and we show that each quasi-invertible element is maximal in R~r with respect to the relation "≤".We consider also a special class of rings,exchange rings,and show that QB-exchange rings are exactly those exchange rings in which every Von Neumann regular element can be extended to a maximal regular element.Finally we study the property of modules which have exchange property,and obtain some good results.We prove that every stronglyπ-regular endomorphism of every module which have exchange property is unit-regular.