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We make a detailed study of idempotent ideals that are traces of countablygenerated projective right modules. We associate to such ideals an ascendingchain of finitely generated left ideals and, dually, a descending chain ofcofinitely generated right ideals.  The study of the first sequence allows us to characterize trace ideals ofprojective modules and to show that projective modules can always be liftedmodulo the trace ideal of a projective module. As a consequence we give somenew classification results of (countably generated) projective modules overparticular classes of semilocal rings. The study of the second sequence leadsus to consider projective modules over noetherian FCR-algebras; we make someconstructions of non-trivial projective modules showing that over such ringsthe behavior of countably generated projective modules that are not direct sumof finitely generated ones is, in general, quite complex.

Reconstructing projective modules from its trace ideal