Proposition 9

— If the group $\mathscr{P}^*$ of principal fractional ideals of K is lattice ordered, then every element x of $K^*$ can be written in the form $x = uv^{-1}$, where u and v are elements of A such that $1 = \gcd(u, v)$; for any other expression $x = u'{v'}^{-1}$ of x as the quotient of two elements of A, we have $u' = uw$ and $v' = vw$, where w is a gcd of $u', v'$; in particular if $1 = \gcd(u', v')$ then u' and v' are associates of u and v respectively.

Such an expression $uv^{-1}$ for an element x of $K^*$ is often called a reduced fraction.