Application of Nagata's Criterion for Factoriality

Definition. A noetherian integral domain $R$ is a Unique Factorization Domain (UFD) if every non-zero non-unit element can be written as a product of prime elements.

Theorem 1 (Nagata’s Criterion for Factoriality). Let $A$ be a noetherian integral domain and let $s\in A$ be a non-zero, non-unit element. If $s$ is a prime element and the localization $A_s$ is a UFD, then $A$ is a UFD.

Theorem 2 (Coordinate Ring of the Real Sphere is a UFD). Let $k=\mathbb{R}$ be the field of real numbers. The ring $A=\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ is a UFD.

Proof. The proof proceeds by applying Nagata’s Criterion to the ring $A$ with a specific choice of a prime element.

First, $A$ is a noetherian integral domain. The ring $\mathbb{R}[x,y,z]$ is noetherian, so its quotient $A$ is noetherian. The polynomial $p(x,y,z)=x^2+y^2+z^2-1$ is irreducible over $\mathbb{R}$, which implies the principal ideal $(p)$ is a prime ideal. Therefore, $A$ is an integral domain.

Let $s=1-z\in A$. The element $s$ is not a unit in $A$, since if it were, $A/(s)$ would be the zero ring. We show $s$ is a prime element by demonstrating that the quotient ring $A/(s)$ is an integral domain.

$$A/(s)=(\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1))/(1-z)\cong \mathbb{R}[x,y,z]/(x^2+y^2+z^2-1,1-z)$$

Substituting $z=1$ into the first relation yields the isomorphism:

$$A/(s)\cong \mathbb{R}[x,y]/(x^2+y^2+1^2-1)\cong \mathbb{R}[x,y]/(x^2+y^2)$$

The polynomial $x^2+y^2$ is irreducible in $\mathbb{R}[x,y]$. Thus, the ideal $(x^2+y^2)$ is a prime ideal, and the quotient ring $\mathbb{R}[x,y]/(x^2+y^2)$ is an integral domain. Consequently, $s=1-z$ is a prime element in $A$.

Next, we show that the localization $A_s=A_{1-z}$ is a UFD. This localization corresponds to the ring of regular functions on the sphere excluding the point $(0,0,1)$. We define an isomorphism $\varphi :\mathbb{R}[u,v]\to A_{1-z}$ induced by the stereographic projection. The map is defined by sending $u$ and $v$ to elements of $A_{1-z}$:

$$\varphi (u)=\frac{x}{1-z},\varphi (v)=\frac{y}{1-z}$$

The inverse map is given by:

$$z=\frac{u^2+v^2-1}{u^2+v^2+1},x=u(1-z)=\frac{2u}{u^2+v^2+1},y=v(1-z)=\frac{2v}{u^2+v^2+1}$$

These establish a biregular correspondence between the punctured sphere and the plane ${\mathbb{R}}^2$, inducing an isomorphism of coordinate rings $A_{1-z}\cong \mathbb{R}[u,v]$.

The ring $\mathbb{R}[u,v]$ is a polynomial ring in two variables over a field, which is a UFD. Therefore, $A_{1-z}$ is a UFD.

Since $A$ is a noetherian integral domain, $s=1-z$ is a prime element in $A$, and the localization $A_s$ is a UFD, Nagata’s Criterion implies that $A$ is a UFD. $\square$