I'm not sure whether this is non-trivial or not, but do there exist simple examples of an affine scheme X having an open affine subscheme U which is not principal in X? By a principal open of X = Spec A, I mean anything of the form D(f) = {P in Spec A : f is not in P}, where f is an element of A.

I'm not sure whether this is non-trivial or not, but do there exist simple examples of an affine scheme $X$ having an open affine subscheme $U$ which is not principal in $X$? By a principal open of $X = \mathrm{Spec} \ A$, I mean anything of the form $D(f) = \{\mathfrak p \in \mathrm{Spec} \ A : f \notin \mathfrak p\}$, where $f$ is an element of $A$.

I'm not sure whether this is non-trivial or not, but do there exist simple examples of an affine scheme X having an open affine subscheme U which is not principal in X?

I'm not sure whether this is non-trivial or not, but do there exist simple examples of an affine scheme X having an open affine subscheme U which is not principal in X? By a principal open of X = Spec A, I mean anything of the form D(f) = {P in Spec A : f is not in P}, where f is an element of A.