Valuation defined on arbitrary commutative ring

Question

My professor introduced the concept of a valuation defined on an arbitrary commutative $A$ ring i.e. a map $v : A \longrightarrow \Gamma \cup \{0\}$ where $\Gamma$ is a totally ordered multiplicative abelian group, such that $v(xy)=v(x)v(y)$, $v(x+y)\le$max$(v(x),v(y))$. He also pointed out that we should introduce the ideal $supp(v):=\{x \in A | v(x)=0 \}$ and take the fraction field of the quotient $A/supp(v)$ to define a ”true valuation”. I apologize if this is not precise, but I ask if somebody could name a book that contains this kind of definitions.

Answer 1

You might like to look at the article:

Rodney Coleman, Laurent Zwald : On Valuation Rings.

You can find it on Internet.

Rodney Coleman