Suppose that $R$ is a commutative ring with no zero divisors. Show that all non zero elements of $R$ have the same additive order.
Attempt: CASE $1$ : When $R$ is finite commutative Ring
Every finite commutative ring with no zero divisors has a unity.
Hence, $\forall~~x \in R,~ n \cdot x = (n \cdot 1) ~x$ where $1$ is the multiplicative identity ( unity)
Hence, the additive order of every element in $R = $ Characteristic of $R$
CASE $2$ : When $R$ is an infinite commutative Ring
In this case, we can't argue that $R$ possesses a unity for sure. Hence, we can't talk about the characteristic as well. How do I proceed ahead ?
Thank you for your help.