Hello to all,

I have been perusing Harthorne for some time, and I noticed something: it is well known that the class group on $\mathbb{P}^n_k$ is $\mathbb{Z}$. But as I look at Harthorne's proof it seems to mee that it works in much greater generality. Namely if I consider any projective scheme $X=Proj(A)$, where $A$ is a graded $UFD$ in such a way that there exists an irreducible element of degree $1$, then the exact same reasoning shows that the class group of $X$ is also $\mathbb{Z}$, and generated by the prime divisor $(a)$. Is this true ?