Is it true that when the integers of a number field are not a UFD then not every point in projective $n$-space over that field can be given by relatively prime algebraic integer coordinates?
When a point can be given by integer coordinates generating a non-principle ideal as GCD then it seems that point cannot be given by relatively prime integers.
But I wonder if I am missing something? Marcello Robbiani "Rational points of bounded height on Del Pezzo surfaces of degree six" at
page 411 seems to say that by taking suitable account of the (finitely many) ideal classes of the ring of integers, every point can be represented with relatively prime integer coordinates. Apparently I am misunderstanding either Robbiani or something about algebraic integers.
Does Robbiani just mean that each point of projective $n$ space can be specified as a relatively-prime-integer $n+1$-tuple times an ideal class?