Prime ideals in $k[x,y]/(xy-1)$.

Question

Let $k$ a field. Let $f$ be the ring injective homomorphism

$$ f:k[x] \rightarrow k[x,y]/(xy-1)$$

obtained as the composition of the inclusion $k[x] \subset k[x,y]$ and the natural projection map $ f:k[x,y] \rightarrow k[x,y]/(xy-1)$.

Prove that there isn't any prime ideal in the ring $k[x,y]/(xy-1)$ whose contraction in $k[x]$ is the prime ideal $(x)$.

Is there any prime ideal in $k[x,y]/(xy-1)$ whose contraction is $(x-1)$?

Thanks! :)

Answer 1

Let $P$ be a prime ideal in $k[x,y]$.

If $xy-1,x\in P$ then $1\in P$, contradiction.

If $xy-1,x-1\in P$ then $y-1\in P$, so we can take $P=(x-1,y-1)$. (Note that $xy-1\in(x-1,y-1)$.)

Answer 2

HINT:

$k[x,y]/(xy-1)$ is naturally isomorphic to the ring of fractions $k[x][\frac{1}{x}] = S^{-1}\ k[x]$, where $S= \{1,x,x^2, \ldots\}$.