Fractional Ideal Inverse

Definition

For any integral domain R with field of fractions F and non-zero fractional ideal I, we define

I1={xF:xIR}.

See the proof that this is an inverse below for some motivation on why we use this definition.

Example

In Z, (aZ)1=1aZ for a0.

Here, aZ is the set of Z multiples of a. The values in F that we can multiply this by in order to stay within the ring are fractions of the form ba for some bZ. That way, the product a×ba=bZ.


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.

Inverse Property

For any integral domain R with non-zero fractional ideal I, we have

I1I=II1=R.
Proof

Our definition for the fractional ideal inverse is very carefully constructed such that we have

I1I={xF:xIR}IR

by definition (stare at the above and think carefully to see how deliberately constructed this is).

Let rR, and iI with i0. Then, we have that riI1 since ri×i=rR. As such, we have that rI1I.

The fact that II1=I1I follows simply from the fact that R is an integral domain, and hence commutative, a property that the ideal multiplication inherits.

Symmetry About Ring

For any integral domain R with fractional ideal I, if IR then I1R.

Proof

First note that R1={xF:xRR}R, and if rR then r{xF:xRR} since 1R, and therefore clearly R1=R (this is just a consequence of the group structure we can construct, but we prove it in a more elementary way here so that this makes sense before that result).

Assume now that IR. If rR, then clearly rF, and rIR since IR and rR, so rI is a set of products of elements in R. As such rI1 and so more generally RI1.

We get the strict inequality by applying R1=R above.