Solution
= Solution
Assume (ii), and let an $R$-submodule $M\subseteq F$ be generated by $x_1,\ldots,x_m$. Choose a common nonzero denominator $d$ and write $x_j=a_j/d$ with $a_j\in R$. Repeated application of (ii) shows that
$$
(a_1,\ldots,a_m)=(g)
$$
for one element $g$: the gcd at each step is a linear combination of the elements processed so far. Therefore
$$
M=R\frac gd,
$$
so (iii) holds. This is the finite-ideal property defining a <Bézout domain>.