On P-exchange rings (Q923687)

A module M over some ring is said to have the exchange property, if for each direct sum \(X=\oplus_{i\in I}X_ i\) of modules containing M as a direct summand there exists a family \(X_ i'\subset X_ i\), \(i\in I\), such that \(M=\oplus_{i\in I}X_ i'\). There is a number of unsolved questions concerning this property, for instance to determine those rings, over which all projective right modules have the exchange property. The present paper contributes some results to this problem. Among other things it is shown that all projective right modules over a ring R have the exchange property if and only if this holds for all Pierce stalks \(R_ x\) of R (for the definition of \(R_ x\) see [\textit{R. S. Pierce}: Modules over commutative regular rings (Mem. Am. Math. Soc. 70, 1967; Zbl 0152.026)]).