When is the formal power series ring a valuation ring?

Question

If $K$ is a field, the formal power series ring in $1$ variable $K[[X]]$ is a discrete valuation ring. What about the many variable case? Is $K[[X_1, \ldots, X_n]]$ a valuation ring?

Instead if we consider a formal power series ring $R[[X_1, \ldots, X_n]]$ where $R$ is a discrete valuation ring, what is known?

It looks like one can define a valuation on $K[[X_1, \ldots, X_n]]$ in the same way as on $K[[X]]$ - as the degree of first non-zero coefficient. Is this correct? — esk, Aug 05 '13 at 10:19
In what ordering? What is the first non-zero coefficient of $x_1 + 2x_2$? — Tobias Kildetoft, Aug 05 '13 at 10:25

score 3 · Answer 1 · answered Aug 05 '13 at 10:58

3

In a valuation ring the ideals are totally ordered by inclusion. What about these two ideals: $(X_1)$ and $(X_2)$?

answered Aug 05 '13 at 10:58

When is the formal power series ring a valuation ring?

1 Answers1