Invertible elements in the ring $K[x,y,z,t]/(xy+zt-1)$ I would like to know how big is the set of invertible elements in the ring $$R=K[x,y,z,t]/(xy+zt-1),$$ where $K$ is any field. In particular whether any invertible element is a (edit: scalar) multiple of $1$, or there is something else. Any help is greatly appreciated.

Let $R=K[X,Y,Z,T]/(XY+ZT-1)$. In the following we denote by $x,y,z,t$ the residue classes of $X,Y,Z,T$ modulo the ideal $(XY+ZT-1)$. Let $f\in R$ invertible. Then its image in $R[x^{-1}]$ is also invertible. But $R[x^{-1}]=K[x,z,t][x^{-1}]$ and $x$, $z$, $t$ are algebraically independent over $K$. Thus $f=cx^n$ with $c\in K$, $c\ne0$, and $n\in\mathbb Z$. Since $R/xR\simeq K[Z,Z^{-1}][Y]$ we get that $x$ is a prime element, and therefore $n=0$. Conclusion: if $f$ is invertible, then $f\in K-\{0\}$.
