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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?

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  • $\begingroup$ What are your thoughts on the question? $\endgroup$ – Vishal Gupta Aug 5 '13 at 8:57
  • $\begingroup$ 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? $\endgroup$ – esk Aug 5 '13 at 10:19
  • $\begingroup$ In what ordering? What is the first non-zero coefficient of $x_1 + 2x_2$? $\endgroup$ – Tobias Kildetoft Aug 5 '13 at 10:25
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In a valuation ring the ideals are totally ordered by inclusion. What about these two ideals: $(X_1)$ and $(X_2)$?

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