Integral Domain with Characteristic Zero

Theorem
In an integral domain with characteristic zero, every non-zero element has infinite order under ring addition.

Proof
Let $\left({D, +, \circ}\right)$ be an integral domain, whose zero is $0_D$ and whose unity is $1_D$, such that $\operatorname{Char} \left({D}\right) = 0$.

Let $x \in D, x \ne 0_D$.

Then:

... that is, $x$ has infinite order in $\left({D, +}\right)$.