Strong lifting

If I is an ideal of a ring R, we say that idempotents lift strongly modulo I if, whenever a2−a∈I, there exists e2=e∈aR (equivalently e2=e∈Ra) such that e−a∈I. The higher socles of R all enjoy this property, as does the Jacobson radical J if idempotents lift modulo J. Many of the useful, basic properties of lifting modulo J are shown to extend to any ideal I with strong lifting, and analogs of the semiperfect and semiregular rings are studied. A number of examples are given that limit possible extensions of the results.