Projective Modules, Idempotent Ideals and Intersection Theorems

We investigate the relationship between projective modules and idempotent ideals for group rings, polynomial rings and more general rings, giving a survey of known results, proving some new results and raising a number of questions. In particular, it is proved that if R is any ring, X a projective right R-module and A an ideal of R such that the R-module X/XA can be generated by a set of elements of cardinality N, for some infinite cardinal N, then X/XB can be generated by a set of elements of cardinality N, where B is the unique maximal idempotent ideal of R contained in A. A recurring theme is that of "intersection theorems" which give information about intersections of powers of ideals of the ring.