Jacobson’s Lemma via Gröbner-Shirshov Bases

Abstract

Let R be a ring with identity 1. Jacobson’s lemma states that for any [Formula: see text], 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, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Gröbner-Shirshov basis theory to obtain the inverse of 1 − ab in terms of (1 − ba)−1, assuming the latter exists.