PlanetMath: fractional ideal of commutative ring

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fractional ideal of commutative ring

(Definition)

Definition. Let be a commutative ring having a regular element and let be the total ring of fractions of . Every -submodule $\mathfrak{a}$ of , generated by a set (not necessarily finite) of elements of , is called fractional ideal of , provided that there exists a regular element of such that $\mathfrak{a}d \subseteq R$ . If the generating set is contained in , the fractional ideal is a usual ideal of , and we can call it an integral ideal of .

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

An ideal $\mathfrak{a}$ (integral or fractional) of is called invertible, if there exists another ideal $\mathfrak{a}^{-1}$ of such that $\mathfrak{aa}^{-1} = R'$ . It is not hard to show that any invertible ideal $\mathfrak{a}$ is finitely generated and regular (i.e., contains a regular element), 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 form 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 .

Note. In the special case that the ring has a unity 1, 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.