I have to prove that the ring $R=K[x,y]/(x^2-y^3+y)$ is not a UFD showing that the prime ideal $(x,y)R$ has height 1, but it's not principal.
Do someone know a simple way to prove it? I know there are others way to solve the problem, for example to consider the Picard group to the elliptic curve, but I am interested to solve it in the way I explained. Thanks!
The ring $K[x,y]$ has Krull dimension $2$. So the ring $R$ has Krull dimension $1$, hence the maximal height of a prime ideal is $1$. And your prime ideal does not have height $0$ because it strictly contains the prime ideal $(x^2-y^3+y)R$.