On Jacobson’s lemma and Drazin invertibility

Abstract

a b s t r a c tIn this note we give an answer to a question of Patricio and da Costa, to note that Jacobson's lemma extends to Drazin invertibility, preserving also the ''Drazin index''. Also, we consider the interaction between abstract versions of Drazin invertibility and Jacobson's lemma.

A class of Drazin inverses in rings

“…In 2009, Patrício and Costa [22] asked if the Drazin invertibility of 1 − ab implies that of 1 − ba. Patrício-Hartwig [24] and Cvetković-Ilić-Harte [8] answered independently this question affirmatively. Lam and Nielsen [19] investigated further Jacobson's Lemma for Drazin inverses, π-regular elements and unit πregular elements in arbitrary rings.…”

Jacobson pairs and Bott-Duffin decompositions in rings

“…Thus, we see that (G) follows from the fact that, for any two elements s, t ∈ R, st is quasi-nilpotent iff ts is quasi-nilpotent; see [7]. (We will actually re-prove this fact as a part of Proposition 3.3 below.)…”

“…This has been done, for instance, in [7] (for Banach algebras). For completeness, we'll include a proof here for general rings.…”

“…There are many generalizations of this magic lemma. For instance, if (α, β) is a Jacobson pair and α is Drazin invertible, then β is also Drazin invertible [7,24]. We say accordingly that Jacobson's Lemma holds for Drazin invertible elements.…”

Jacobson’s Lemma via Gröbner-Shirshov Bases

“…Jacobson's lemma [9,4] states that for any a, b ∈ R, if 1 − ab is invertible then so is 1 − ba. Jacobson's lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse [2,6], generalized Drazin inverse [11], and inner inverse [3].…”