invertible ideals are projectiveInvertibleIdealsAreProjective2008-12-07 20:40:222008-12-07 20:44:15Theoremequivalent characterizations of Dedekind domainsprojective modulefractional idealinvertible ideal% this is the default PlanetMath preamble. as your knowledge
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If $R$ is a ring and $f\colon M\rightarrow N$ is a homomorphism of $R$-modules, then a right inverse of $f$ is a homomorphism $g\colon N\rightarrow M$ 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.
\begin{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.
\end{theorem*}
In particular, every fractional ideal over a Dedekind domain is invertible, and is therefore projective.