Abstract
If I is an ideal of a ring R, we say that idempotents lift strongly modulo I if, whenever a 2 − a ∈ I , there exists e 2 = e ∈ a R (equivalently e 2 = e ∈ R a ) 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.
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