Lifting Projectives

Abstract

Let R be a ring with radical (all rings have a unit element, all modules are unital). Often, one wishes to lift modules modulo , that is, to a given, say, left R / -module U find a left R -module E with the property that E / E ≃ U . This is of course not always possible. Here I prove, roughly, that if a finitely generated projective U can be lifted at all, it can be lifted to a projective. Or rather, if U can be lifted to an E satisfying a certain mild condition, then E is projective (Lemma).