Nilpotent Ring Element plus Unity is Unit

Theorem
Let $A$ be a ring with unity.

Let $1 \in A$ be its unity.

Let $a \in A$ be nilpotent.

Then $1 + a$ is invertible.

Proof
Because $a$ is nilpotent, there exists a natural number $n > 0$ with $a^n = 0$.

By Sum of Geometric Progression in Ring:
 * $(1 + a) \cdot \displaystyle \sum_{k \mathop = 0}^{n-1} (-a)^k = 1 + (-a)^n$
 * $\left( \displaystyle \sum_{k \mathop = 0}^{n-1} (-a)^k \right) \cdot (1 + a) = 1 + (-a)^n$

where $\sum$ denotes summation.

By Negative of Nilpotent Ring Element, $(-a)^n = 0$.

Thus $1 + a$ is a unit.

Generalizations

 * Nilpotent Ring Element plus Unit is Unit
 * Topologically Nilpotent Element plus Unity is Invertible