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:

\(\ds \forall n \in \Z_{>0}: \, \) \(\ds n \cdot x\) \(=\) \(\ds n \cdot \paren {x \circ 1_D}\)
\(\ds \) \(=\) \(\ds \paren {n \circ 1_D} \cdot x\) Integral Multiple of Ring Element


Then:

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


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

$\blacksquare$


Sources