Definitions of valuations in terms of totally ordered group

Question

Wikipedia gives a definition of valuations involving abelian totally ordered groups. So far I have only seen valuations taking values in the real numbers. Is there a reason for this generalization?

Answer 1

Ultimately it doesn't really matter too much, note that such totally ordered groups cannot have torsion, for if the order of some element, $x$, is $m>0$ then $(m-1)x=-x$ so that if $x>0$ we have

$$(m-1)x=\underbrace{x+\ldots +x}_{(m-1)\text{ times}}>0$$

however $-x<0$ since $x+-x=0$ a contradiction. We conclude that each element has infinite order, so cyclic subgroups, you may as well think of the valuation as taking place in $\Bbb Z$.

Provided with some relatively mild hypotheses, you can similarly embed $(\Gamma, +, \le)$ into $(\Bbb R, +,\le)$ in a way that preserves the ordering. The formal approach is taken because the only properties about $\Bbb R$ we use are that it is an abelian, totally-ordered group. It's the same reason why many theorems on $\Bbb Z$ are abstracted to PIDs or Dedekind domains or UFDs: people want to see the most general setting in which they can phrase the theorems so we can learn more about the essential, underlying structures involved.

In particular, it is enough to have the group $(\Gamma, +,\le)$ in order to define a valuation topology on the ring in question, and this concept is very fruitful in proving algebraic theorems about the ring, much in the same way one can prove there are infinitely many primes by appropriately topologizing the integers (even though the former fact is, at a glance, purely algebraic).

If it bothers you, I recommend you just ignore it. The cases of prime interest are when the valuation is discrete, and a moderately distant second is $\Gamma=\Bbb R$. The more arcane/abstract cases are usually only of interest to those who specifically study things in that degree of generality.

Answer 2

I know this is an old question but I just want to chime in with the fact that I really disagree with the answer posted here. Valuations with image that can be take in the reals are called valuations of rank $1$, and "higher rank" valuations play an important role in the theory of adic spaces, which plays an important role in Scholze's theory of perfectoid spaces, which is an important topic in non-archimedean geometry and, more generally, arithmetic geometry at the moment.