I am reading a paper for a summer research project (Example of an interpolation domain ). I am unfamiliar with some of the terms used here and I have tried searching on google for definitions but I am a little confused. I am not sure if the definitions I am finding pertain to what I am looking at. The passage that I am at states:
... we take $v(r)$ to be the smallest exponent on a nonzero term of $r$. Denote the valuation ring of $v$ by $V$; so that, to be in $V$, a series must have no (nonzero) terms with negative exponents. Because $V$ has value group $G$, its maximal ideal is idempotent, and hence ...
(The quote comes from the bottom of the first paragraph in Example 1 on page 2.)
We are doing long division of polynomials and writing them as a kind of Laurent series and each non-zero element of the fraction field has a unique kind of "Laurent expansion." I have worked a little bit with valuations before but in what seemed to be kind of informal as the definition that I worked with then I have not seen repeated in my searches recently. Specifically, I worked with the $p$-adic valuation on $\mathbb{Z}$ defined by $v_p(x)=p^{-n}$ where $x=p^nb$, $p\nmid b$ for $x\neq 0$ and $v_p(x)=0$ if $x=0$.
I just want to learn more about valuations, valuation rings, value groups, and an ideal being idempotent. From what I gather, you define a valuation on a field and then this gives rise to a valuation ring. The definitions of valuation that I have looked at involve $\infty$ which confuses me a little bit as it looks like each non-zero element gets mapped to a rational number in this case, as the exponents are rational. Looking at when I worked with the $p$-adic valuation on $\mathbb{Z}$, we never mapped $0 \mapsto \infty$.
Maybe there is material that is available online I can look at to work through elementary results with valuations and valuation rings. Any advice on places I can go to find more information about these things? Thanks very much.