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Let $\kappa$ be a field and $S=\kappa[[X]]$ be the ring of power series which depends on the indeterminate $X$. Now consider the ring $S[Y]$, the ring of polynomials with coefficients in $S$ and indeterminate $Y$.

$\Bbb{Q}$usetion: How can we could prove that the ideal $(XY-1)$ in $S[Y]$ is a prime ideal of height 1?

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  • $\begingroup$ It might be easier solve the problem by replacing $\kappa[[X]]$ by a discrete valuation ring and $X$ by a uniformizing element. $\endgroup$
    – Cantlog
    Dec 19, 2013 at 16:49

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Let $A$ be a discrete valuation ring. Then $A_\pi$ is a field where $\pi$ is a uniformizing parameter; this follows immediately from the fact that every element in a DVR is unit times a non-negative power of the uniformizing parameter.

Thus $A[Y]/(\pi Y - 1)$ is a field and so $\pi Y - 1$ is maximal.

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