Fractional ideals

Neither ideal multiplication nor submodule multiplication have multiplicative inverses. Indeed, ideal multiplication forms a commutative H-trivial semigroup. Yet, in spite of this we often talk of fractional ideals: a related semigroup on a set system in which certain sets have inverses. It is clear that in order to construct inverses we need a different sort of structure then a module or a ring. Instead we should turn to R-algebras.

Definition. let $R$ be a subring of another ring $K$, then $\frac{K}{R}$ forms an extension R-algebra with addition and multiplication provided by $K$ and scalar multiplication provided by the subring $R$.

Example. let $R$ be an integral domain and let $K$ be its field of fractions then $\frac{K}{R}$ forms an extension R-algebra.

The point is that now unlike with rings and modules, we have two different types of multiplication to take care of: (1) scalar multiplication by the subring elements and (2) the multiplication operation of the R-algebra. In both submodules and ideals the multiplication of sets was the same multiplication used to define the sets. By having two different kinds of multiplication, we make possible inverses and fractions, which is a significant difference then with ideal multiplication.

The definition of fractional ideals of the field of fractions extension R-algebra of an integral domain proceeds with a submodule I closed under addition and scalar multiplication for which there is a number $d$ such that $dI \subset R$. The lattice operations are inherited from submodules, but here fractional ideal multiplication is defined by the ordinary multiplication operation of the R-algebra, rather then the scalar multiplication used to define fractional ideals themselves.

Multiplication of fractional ideals forms a semigroup, as for any two ideals $I,J$ then the numbers $d_I, d_J$ can be multiplied to get $d_I d_J$ which clears out the denominators in $IJ$. Unlike the ideal multiplication semigroup, which is commutative and group-free these semigroups have much more varied behavior which is related to comparison to the identity element:

• The full set of scalars $R$ is the identity element because it is neither increasing nor decreasing
• Subidentity ideals are decreasing as a special case of multiplication by submodules
• Superidentity ideals are increasing because they contain $1$ which is a multiplicative identity

The elements that are incomparable to $R$ have different and varied behavior. This shows that for fractional ideals, multiplication can be either increasing/decreasing and so it clearly forms a different kind of semigroup then ideal multiplication. Fractional ideals can even have inverses, and the set of all invertible fractional ideals, the group of units, forms a subgroup of the fractional ideal semigroup.

In the case of a Dedekind domain, we further know that all non-zero ideals have inverses. This means we can go from a setting with no multiplicative inverses to ones with all of them. Much like how the commutative semigroup $(\mathbb{N},*)$ has no inverses but all of its non-zero elements do in $(\mathbb{Q}_+,*)$ for a Dedekind domain, non-zero ideals have no inverses but once they are embedded in the fractional ideal semigroup they all do. So for a Dedekind domain, the introduction of fractional ideals is like the introduction of fractions in number theory.