Invertible plus nilpotent implies invertible

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