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 $\struct {D, +, \circ}$ be an integral domain, whose zero is $0_D$ and whose unity is $1_D$, such that $\Char D = 0$.

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

Then:

Then:

That is, $x$ has infinite order in $\struct {D, +}$.