fractional ideal of commutative ring

Definition. Let $R$ be a commutative ring having a regular element and let $T$ be the total ring of fractions of $R$ . An $R$ -submodule $\mathfrak{a}$ of $T$ is called fractional ideal of $R$ , provided that there exists a regular element $d$ of $R$ such that $\mathfrak{a}d \subseteq R$ . If a fractional ideal is contained in $R$ , it is a usual ideal of $R$ , and we can call it an integral ideal of $R$ .

Note that a fractional ideal of $R$ is not necessarily a subring of $T$ . The set of all fractional ideals of $R$ form under the multiplication an commutative semigroup with identity element $R' = R\!+\!\mathbb{Z}e$ , where $e$ is the unity of $T$ .

An ideal $\mathfrak{a}$ (integral or fractional) of $R$ is called invertible, if there exists another ideal $\mathfrak{a}^{-1}$ of $R$ such that $\mathfrak{aa}^{-1} = R'$ . It is not hard to show that any invertible ideal $\mathfrak{a}$ is finitely generated and regular, moreover that the inverse ideal $\mathfrak{a}^{-1}$ is uniquely determined (see the entry ``invertible ideal is finitely generated'') and may be generated by the same amount of generators as $\mathfrak{a}$ .

The set of all invertible fractional ideals of $R$ forms an Abelian group under the multiplication. This group has a normal subgroup consisting of all regular principal fractional ideals; the corresponding factor group is called the class group of the ring $R$ .

Note. In the special case that the ring $R$ has a unity 1, $R$ itself is the principal ideal (1), being the identity element of the semigroup of fractional ideals and the group of invertible fractional ideals. It is called the unit ideal. The unit ideal is the only integral ideal containing units of the ring.