Unity plus Negative of Nilpotent Ring Element is Unit

Theorem
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $x \in R$ be nilpotent.

Then $1_R - x$ is a unit of $R$.

Proof
By definition of nilpotent element:


 * $x^n = 0_R$

for some $n \in \Z_{>0}$.

From Difference of Two Powers:

for $a, b \in R$.

Putting $a = 1_R$ and $b = x$, we have:

Thus by definition $1_R - x$ has a product inverse $1_R + x + x^2 + \dotsb + x^{n - 2} + x^{n - 1}$.

Hence by definition $1_R - x$ is a unit of $R$.