invertible ideals are projective

If R is a ring and f:MN is a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of R-modules, then a right inverseMathworldPlanetmathPlanetmath of f is a homomorphism g:NM such that fg is the identity map on N. For a right inverse to exist, it is clear that f must be an epimorphismMathworldPlanetmathPlanetmath. If a right inverse exists for every such epimorphism and all modules M, then N is said to be a projective moduleMathworldPlanetmath.

For fractional idealsMathworldPlanetmathPlanetmath over an integral domainMathworldPlanetmath R, the property of being projective as an R-module is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to being an invertible ideal.


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 domainMathworldPlanetmath 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