Fractional Ideal Inverse
For any integral domain
See the proof that this is an inverse below for some motivation on why we use this definition.
In
Here,
From this we establish some important basic properties of the inverse which are essential for it being used as a multiplicative inverse in the fractional ideal group.
For any integral domain
Proof
Our definition for the fractional ideal inverse is very carefully constructed such that we have
by definition (stare at the above and think carefully to see how deliberately constructed this is).
Let
The fact that
For any integral domain
Proof
First note that
Assume now that
We get the strict inequality by applying