# Invertible plus nilpotent implies invertible

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## Statement

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

## Facts used

$1 + x^n = (1 + x)(1 - x + x^2 - \ldots + (-1)^{n-1}x^{n-1})$

## Proof

Given: A commutative unital ring $A$, elements $a,x \in A$ such that $a$ is invertible, and $x^n = 0$ for some positive integer $n$

To prove: $a + x$ is invertible

Proof: Since $a$ is invertible, it suffices to prove that $1 + x/a$ is invertible. Since $x^n = 0$, we also have $(x/a)^n = 0$. The above formula then tells us that $1 + x/a$ is invertible, completing the proof.