fractional ideal of commutative ring

Definition.  Let R be a commutative ring having a regular elementPlanetmathPlanetmathPlanetmath and let T be the total ring of fractionsMathworldPlanetmath of R.  An R-submodule ( π”ž of T is called fractional idealMathworldPlanetmathPlanetmath of R, provided that there exists a regular element d of R such that  π”žβ’dβŠ†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 elementMathworldPlanetmath  Rβ€²=R+℀⁒e,  where e is the unity of T.

An ideal π”ž ( or fractional) of R is called invertible, if there exists another ideal π”ž-1 of R such that  π”žβ’π”ž-1=Rβ€².  It is not hard to show that any invertible ideal π”ž is finitely generatedMathworldPlanetmathPlanetmath and regular (, moreover that the inverse ideal π”ž-1 is uniquely determined (see the entry β€œinvertible ideal is finitely generated (”) and may be generated by the same amount of generatorsPlanetmathPlanetmath ( as π”ž.

The set of all invertible fractional ideals of R forms an Abelian groupMathworldPlanetmath under the multiplication.  This group has a normal subgroupMathworldPlanetmath 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 idealMathworldPlanetmathPlanetmath (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 groupMathworldPlanetmath of a ring
Defines unit ideal