What is a "normalized valuation" corresponding to a valuation ring? I encountered the phrase "normalized valuation" similar to the following:

Let $A_i$ be the valuation ring $k[x_1,...,x_n]_{\langle x_i\rangle}$ and $v_i$ be the normalized valuation defined by $A_i$.

I didn't know this term before, and a short internet search did not help me.
What I know: we can define a map $k[x_1,...,x_n]\smallsetminus\{0\}\to\mathbb{Z}$ by sending $f=gx_i^{n_f}$ with $x_i\nmid g$ to $n_f\in\mathbb{Z}$. Then extend this to $v:Q(k[x_1,...,x_n])^*=k(x_1,...,x_n)^*\to\mathbb{Z}$ via $\frac{f}{g}\mapsto n_f-n_g$, and this is a discrete valuation on $k(x_1,...,x_n)$ with $k[x_1,...,x_n]_v=k[x_1,...,x_n]_{\langle x_i\rangle}$ as its discrete valuation ring. Is the map $v$ already the $v_i$ mentioned above? Is it "normalized", and what does this mean?
Also, the definition of the map $v$ relies on $k[x_1,...,x_n]$ being a UFD. So by the same argument, $R_{\langle p\rangle}$ is a DVR if $R$ is a UFD and $p\in R$ prime. So I guess this does no longer hold for rings of the form $R_\mathfrak{p}$ where $\mathfrak{p}\subset R$ is prime in general? What about $k[x_1,...,x_n]_{\langle x_1,x_2\rangle}$?
Thank you!
 A: A normalized discrete valuation $v$ of a field $K$ means that $v$ is a discrete valuation of $K$ such that $v(K^*) = \mathbb{Z}$. In general, $v(K^*)$ can be any discrete subgroup of $\mathbb{R}$.
A: For general prime ideals $\mathfrak{p}\subseteq k[x_1,\ldots, x_n]$, the local ring $k[x_1,\ldots, x_n]_\mathfrak{p}$ will not be a valuation ring, and hence will not define a valuation. But that does not mean you cannot consider valuations on such a ring; in fact, it is a very interesting thing to do!
Take, for instance, the local ring $R = \mathbb{C}[x,y]_{(x,y)}$. This is not a valuation ring, but it makes sense to consider valuations $\nu\colon R\to \mathbb{R}\cup\{+\infty\}$. In fact, a rather interesting book (The Valuative Tree by Favre and Jonsson) has been written about valuations $\nu\colon R\to \mathbb{R}\cup\{+\infty\}$ satisfying the properties: 


*

*$\nu$ only takes nonnegative values on $R$.

*$\nu(z) = 0$ for all $z\in \mathbb{C}^\times$.

*$\nu(f)>0$ for all $f$ in the maximal ideal $\mathfrak{m}$ of $R$.


Such $\nu$ are called centered valuations. Such valuations also have a notion of normalized. We say a centered valuation $\nu$ is normalized if $\min_{f\in \mathfrak{m}} \nu(f) = 1$. It turns out that the set of normalized centered valuations on $R$ has an interesting combinatorial structure: it is an $\mathbb{R}$-tree. 
The study of spaces of valuations on rings is becoming a more popular subject these days, sometimes falling under the heading of nonarchimedean analytic geometry or Berkovich geometry. 
