Integral Domain with Characteristic Zero

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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:

\(\displaystyle \forall n \in \Z_{>0}: \ \ \) \(\displaystyle n \cdot x\) \(=\) \(\displaystyle n \cdot \paren {x \circ 1_D}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {n \circ 1_D} \cdot x\) Powers of Ring Elements


Then:

\(\displaystyle x\) \(\ne\) \(\displaystyle 0_D\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle n \cdot 1_D\) \(\ne\) \(\displaystyle 0_D\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle n \cdot x\) \(\ne\) \(\displaystyle 0_D\) Definition of Integral Domain


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

$\blacksquare$


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