Idempotent lifting and ring extensions

摘要

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring R for which idempotents lift modulo the Jacobson radical J(R), but idempotents do not lift modulo J(M-2(R)). Thus, the property "idempotents lift modulo the Jacobson radical" is not a Morita invariant. We also prove that if I and J are ideals of R for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over I + J. On the positive side, if I and J are enabling ideals in R, then I + J is also an enabling ideal. We show that if I (sic) R is (weakly) enabling in R, then I[t] is not necessarily (weakly) enabling in R[t] while I [t] is (weakly) enabling in R[t]. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.