Thanks for your answers!!

If you take $A=\mathcal{O}_K$, the invertible $A$-modules are exactly the non-zero ideals of $A$, so I guess we can reformulate the exercise to be

There is a bijection of sets
$\{(n+1)$-tuples of elements of A such that $\exists i: a_i\neq 0 \}$ modulo equivalence, where equivalence is multiplication by a unit

and 

$\{A$-valued points of $\mathbb{P}^n_A\}$

Thus, for $\mathcal{O}_K$, the "classical" definition of points of projective $n$-space coincides with the definition of $\mathcal{O}_K$-valued points of $\mathbb{P}^n_{\mathcal{O}_K}$.