 Nonprincipal coordinate ideal in two variables

Over a field $F$, the ideal $(X,Y)$ in a two-variable polynomial ring is not principal. A proposed generator would divide both $X$ and $Y$. Any nonconstant divisor of the degree-one polynomial $X$ is a scalar multiple of $X$, which does not divide $Y$; therefore the generator would have to be a unit. But every element of $(X,Y)$ vanishes at $(0,0)$, so the ideal cannot contain one. This contrasts with $F[X]$, where the division algorithm makes every ideal principal.