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单词 InvertibleIdealsAreProjective
释义

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.

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 domainMathworldPlanetmath is invertible, and is therefore projective.

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更新时间:2025/5/4 21:42:00