Fractional ideal of a Dedekind domain A is a projective module

Question

It can be known that any fractional ideal $\alpha$ of A can be generated by two elements, and is that enough to construct a reverse of the quotient map from $A^2$ to $\alpha$? And by the way, I wonder whether Dedekind domain is a UFD, and does g.c.d of two elements makes sense.

Answer 1

A Dedekind domain is not necessarily a U.F.D. If it is, it actually is a P.I.D. For fractional ideals, a theorem asserts that, over a Dedekind domain, any finitely generated torsion-free module is projective.