Let $\mathfrak{o}$ be an integral domain and $K$ its field of fractions.
By a fractional ideal of $\mathfrak{o}$ we mean an $\mathfrak{o}$-submodule $\mathfrak{A}$ of $K$ such that $$z\mathfrak{o} \subseteq \mathfrak{A} \subseteq u\mathfrak{o} \mbox{ for some } z,u\in K^{\times}. \hskip1cm (*)$$ We note that (*) certainly holds when $0\neq \mathfrak{A}\subseteq \mathfrak{o}$.
I do not understnad the last line above. This is from P. M. Cohn's Basic algebra book.
If $0\neq \mathfrak{A}\subseteq \mathfrak{o}$ then take $u=1$ so that in (*) the second inequality holds; what about first?
This may be trivial, but I didn't get it; some books mention that $\mathfrak{A}$ is finitely generated $\mathfrak{o}$-submodule of $K$ in the definition of fractional ideal; but it was not the case in Cohn's book.