Fractional ideal

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.

Definition and basic results

Let

R

{\displaystyle R}

be an integral domain, and let

K

=

Frac

⁡

R

{\displaystyle K=\operatorname {Frac} R}

be its field of fractions.

A fractional ideal of

R

{\displaystyle R}

is an

R

{\displaystyle R}

-submodule

I

{\displaystyle I}

of

K

{\displaystyle K}

such that there exists a non-zero

r

∈

R

{\displaystyle r\in R}

such that

r

I

⊆

R

{\displaystyle rI\subseteq R}

. Equivalently,

I

⊆

K

{\displaystyle I\subseteq K}

is a fractional ideal of

R

{\displaystyle R}

if

I

=

r

−

1

J

{\displaystyle I=r^{-1}J}

, where

r

{\displaystyle r}

is a non-zero element of

R

{\displaystyle R}

and

J

{\displaystyle J}

is an ideal of

R

{\displaystyle R}

. The element

r

{\displaystyle r}

can be thought of as clearing out the denominators in

I

{\displaystyle I}

, hence the name fractional ideal.

The principal fractional ideals are those

R

{\displaystyle R}

-submodules of

K

{\displaystyle K}

generated by a single nonzero element of

K

{\displaystyle K}

. A fractional ideal

I

{\displaystyle I}

is contained in

R

{\displaystyle R}

if and only if it is an (integral) ideal of

R

{\displaystyle R}

.

A fractional ideal

I

{\displaystyle I}

is called invertible if there is another fractional ideal

J

{\displaystyle J}

such that

I

J

=

R

{\displaystyle IJ=R}

where

I

J

=

{

a

1

b

1

+

a

2

b

2

+

⋯

+

a

n

b

n

:

a

i

∈

I

,

b

j

∈

J

,

n

∈

Z

>

0

}

{\displaystyle IJ=\{a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}:a_{i}\in I,b_{j}\in J,n\in \mathbb {Z} _{>0}\}}

is the product of the two fractional ideals.