If $R$ is $\text{UFD},$ then $R[X,Y]$ is $\text{UFD}.$

Question

Let $R$ be commutative ring with $1.$ Suppose $R$ is $\text{UFD}.$ Could anyone advise me on how to prove $R[X,Y]$ is $\text{UFD}\ ?$ Thank you.

Answer 1

If $R$ is a UFD then $R[x]$ is a UFD, see any textbook in algebra. Now set $S=R[x]$ and consider $S[y]=R[x,y]$. Also, the Laurent ring $R[x,x^{-1}]$ is again a UFD.