# Invertible plus nilpotent implies invertible

From Commalg

This article is about the statement of a simple but indispensable lemma in commutative algebra

View other indispensable lemmata

## Statement

In a commutative unital ring, the sum of an invertible element and a nilpotent element is an invertible element.

## Facts used

## Proof

*Given*: A commutative unital ring , elements such that is invertible, and for some positive integer

*To prove*: is invertible

*Proof*: Since is invertible, it suffices to prove that is invertible. Since , we also have . The above formula then tells us that is invertible, completing the proof.