Solution
= Solution
Assume (iii). Because $R$ is a <Noetherian ring>, every <ideal> $I\subseteq R$ is finitely generated. Regard $I$ as an $R$-submodule of $F$. By (iii), $I=Rx$ for some $x\in F$; since $x\in I\subseteq R$ unless $I=0$, this is a principal <ideal> of $R$. Thus every <ideal> is principal and (i) holds.
Together with the previous two implications, this proves (i), (ii), and (iii) equivalent.