fractional ideal

Last updated January 27, 2022

Definition

Let $\mathcal{O}$ be an integral domain, $k$ its field of fractions.

An $\mathcal{O}$-submodule $I\subset k$ is called a fractional ideal iff $I\neq {0}$ and $I^{-1}\neq 0$.

Examples

1. Let $\mathcal{O}$ be a Noetherian integrally closed domain. Then $I\subset k$ is a fractional ideal iff $I$ is nonzero and finitely generated.
   1. Then $I^{-1}$ is a f.i.
   2. $\mathcal{O}(I)=\mathcal{O}$

Properties

1. Let $\mathcal{O}$ be a NICD. Then
   1. If $I$ is a f.i. so is $I^{-1}$
   2. $\mathcal{O}(I)=\mathcal{O}$
   3. If $I, J$ are f.i.’s then so are $I+J$, $IJ$, and $I\cap J$.