modules over infinite fields (and rings?): does every nonzero element have infinite order?

Question

on p359 of dummit/foote (intro paragraph to tensor products of modules), it says

every nonzero element in a vector space over $Q$ has infinite (additive) order

i'm not sure why this is obvious?

Answer 1

To expand on the comments of Lord Shark and Joppy: if $V$ is a vector space over $\Bbb Q$ and

$v \in V \tag 1$

has finite additive order $n$, so that $n$ $v$s added together yield $0$,

$\displaystyle \sum_1^n v = 0, \tag 2$

then

$nv = \displaystyle \sum_1^n v = 0; \tag 3$

now since

$n^{-1} \in \Bbb Q, \tag 4$

we have

$v = 1v = (n^{-1}n)v$ $= n^{-1}(nv) = n^{-1}(0) = 0; \tag 5$

thus if

$v \ne 0, \tag 6$

the additive order of $v$ cannot be finite.