Ring of polynomial functions on unit hyperbola is PID

Question

Let $R=\mathbb{R}[X,Y]/(XY-1)$ be the ring of polynomial functions pn the unit hyperbola. How do I prove that $R$ is a principal ideal domain with unit group $$R^*=\{cX^i\text{mod}(XY-1):c\in\mathbb{R}^*,i\in\mathbb{Z}\}\cong\mathbb{R}^*\times<X>?$$

I know that $R$ is isomorphic with the ring of Laurent polynomials, but how can I use that?

Answer 1

A principal ideal domain is a ring where every nonzero prime ideal has height one. First notice that $(XY-1)$ is a prime ideal. And any prime ideal $p$ of the ring $\frac{\mathbb{R}[X,Y]}{(XY-1)}$ is in one to one correspondence to the prime ideals of the ring $\mathbb{R}[X,Y]$ containing $(XY-1)$. Now as the dimension of $\mathbb{R}[X,Y]$ is 2 so any nonzero prime ideal of $\frac{\mathbb{R}[X,Y]}{(XY-1)}$ has height 1, hence it is a PID.

Answer 2

Surely you know that $\mathbb R[X]$ is a PID. Now your ring is equal to $\mathbb R[X,X^{-1}]$ (sitting inside $\mathbb R(X)$). It is an easy exercise to prove that if $A$ is a PID and $a$ is a non-zero element of $A$, then $A[1/a]$ is a PID.

As for the unit group, what are the units in $\mathbb R[X]$? Which additional elements become invertible when you invert $X$?

(You could consider the analogy with $\mathbb Z$ and $\mathbb Z[1/2]$.)