invertible ideals are projective
If is a ring and is a homomorphism of -modules, then a right inverse
of is a homomorphism such that is the identity map on . For a right inverse to exist, it is clear that must be an epimorphism
. If a right inverse exists for every such epimorphism and all modules , then is said to be a projective module
.
For fractional ideals over an integral domain
, the property of being projective as an -module is equivalent
to being an invertible ideal.
Theorem.
Let be an integral domain. Then a fractional ideal over is invertible if and only if it is projective as an -module.
In particular, every fractional ideal over a Dedekind domain is invertible, and is therefore projective.