Idempotent lifting and ring extensions

Abstract

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 [Formula: see text] for which idempotents lift modulo the Jacobson radical [Formula: see text], but idempotents do not lift modulo [Formula: see text]. Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if [Formula: see text] and [Formula: see text] are ideals of [Formula: see text] for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over [Formula: see text]. On the positive side, if [Formula: see text] and [Formula: see text] are enabling ideals in [Formula: see text], then [Formula: see text] is also an enabling ideal. We show that if [Formula: see text] is (weakly) enabling in [Formula: see text], then [Formula: see text] is not necessarily (weakly) enabling in [Formula: see text] while [Formula: see text] is (weakly) enabling in [Formula: see text]. 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.