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 (http://planetmath.org/Submodule) $\mathfrak{a}$ of $T$ is called 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  $R^{\prime}=R\!+\!\mathbb{Z}e$,  where $e$ is the unity of $T$.

An ideal $\mathfrak{a}$ ( or fractional) of $R$ is called invertible, if there exists another ideal $\mathfrak{a}^{-1}$ of $R$ such that  $\mathfrak{aa}^{-1}=R^{\prime}$.  It is not hard to show that any invertible ideal $\mathfrak{a}$ is finitely generated   and regular (http://planetmath.org/RegularIdeal), moreover that the inverse ideal $\mathfrak{a}^{-1}$ is uniquely determined (see the entry “invertible ideal is finitely generated (http://planetmath.org/InvertibleIdealIsFinitelyGenerated)”) and may be generated by the same amount of generators  (http://planetmath.org/GeneratorsOfInverseIdeal) 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 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.

 Title fractional ideal of commutative ring Canonical name FractionalIdealOfCommutativeRing Date of creation 2015-05-06 14:40:32 Last modified on 2015-05-06 14:40:32 Owner pahio (2872) Last modified by pahio (2872) Numerical id 16 Author pahio (2872) Entry type Definition Classification msc 13B30 Related topic FractionalIdeal Related topic GeneratorsOfInverseIdeal Related topic IdealClassesFormAnAbelianGroup Defines fractional ideal Defines integral ideal Defines invertible ideal Defines invertible Defines inverse ideal Defines class group  of a ring Defines unit ideal