invertible ideals are projective

If $R$ is a ring and $f:M\to N$ is a homomorphism of $R$-modules, then a right inverse of $f$ is a homomorphism $g:N\to M$ such that $f\circ g$ is the identity map on $N$. For a right inverse to exist, it is clear that $f$ must be an epimorphism. If a right inverse exists for every such epimorphism and all modules $M$, then $N$ is said to be a projective module.

For fractional ideals over an integral domain $R$, the property of being projective as an $R$-module is equivalent to being an invertible ideal.

Theorem.

Let $R$ be an integral domain. Then a fractional ideal over $R$ is invertible if and only if it is projective as an $R$-module.

In particular, every fractional ideal over a Dedekind domain is invertible, and is therefore projective.

Title	invertible ideals are projective
Canonical name	InvertibleIdealsAreProjective
Date of creation	2013-03-22 18:35:47
Last modified on	2013-03-22 18:35:47
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	5
Author	gel (22282)
Entry type	Theorem
Classification	msc 16D40
Classification	msc 13A15
Related topic	ProjectiveModule
Related topic	FractionalIdeal