# Suppose that R is a commutative ring without zero-divisors. Show that all the nonzero elements of R have the same additive order.

Suppose that R is a commutative ring without zero-divisors. Show that all the nonzero elements of R have the same additive order.
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Ayesha Gomez

Consider p and q be any nonzero element of R.
therefore,
$pq+pq+pq+...n×=n\cdot \left(pq\right)$
$=\left(n×p\right)×q$
$=p\left(n×q\right)$
Since,
$n×p=0$
$\left(n×p\right)q=0$
$p\left(n×q\right)=0$
With no zero divisiors,
$n×q=0$, therefore,
and if the $pq+pq+pq+...m×=m\cdot \left(pq\right)$
$=\left(m\cdot p\right)\cdot q$
$=p\left(m\cdot q\right)$
and, $m\cdot q=0$
$\left(m\cdot p\right)\cdot q=0$
$p\left(m\cdot q\right)=0$
with no zero divisors, , $m\cdot q=0$
Therefore, $n\le m\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}m\le n$