# Jacobson's lemma via Groebner-Shirshov bases

## TL;DR

This paper presents a constructive method using Groebner-Shirshov bases to derive the inverse of 1 - ab from the inverse of 1 - ba in rings, extending Jacobson's lemma to generalized inverses.

## Contribution

It introduces a new algebraic approach to explicitly compute inverses related to Jacobson's lemma using Groebner-Shirshov basis theory.

## Key findings

- Provides a constructive algebraic procedure for inverse calculation
- Extends Jacobson's lemma to generalized inverses
- Offers explicit formulas for inverses in ring theory

## Abstract

Let $R$ be a ring with identity $1$. Jacobson's lemma states that for any $a,b\in R$, 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, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Groebner-Shirshov basis theory to obtain the inverse of $1-ab$ in terms of $(1-ba)^{-1}$, assuming the latter exists.