show $\mathbb{C} [x,y]/(xy-1)$ is a principal ideal domain

Asked May 17 '19 at 17:44

Active May 18 '19 at 08:31

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I've got to show that $A:=\mathbb{C} [x,y]/(xy-1)$ is a principal ideal domain

I know that $A$ is isomorphic to $\mathbb{C}[t,t^{-1}]$ and that this a subfield of $\mathbb{C}[t]$ which is a PID. So then $A$ is also a PID.

But however I've got to work with the canonical inclusion map $i: \mathbb{C}[x] \hookrightarrow A$ in order to show then $I \cap \operatorname{Im}(i) \neq \{0\}$ for every ideal $I \neq \{0\}$. I have no idea why I need this.

edited May 18 '19 at 08:31

asked May 17 '19 at 17:44

Thesinus

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$A$ is not a subfield of $\mathbb{C}[t]$, because $A$ is not a subset of $\mathbb{C}[t]$ and neither $A$ nor $\mathbb{C}[t]$ are fields (check the inverse of $t+1$). – Aphelli May 17 '19 at 17:55
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Also, being a subring of a PID does not automatically make it a PID. – rschwieb May 17 '19 at 18:01
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As you noted, it's isomorphic to $\mathbb C[t,t^{-1}]$, which is the localization of $\mathbb C[t]$ at the powers of $t$. Then a nonzero localization of a PID is a PID – rschwieb May 17 '19 at 18:03

show $\mathbb{C} [x,y]/(xy-1)$ is a principal ideal domain

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