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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.