Definition
If R is an integral domain, define a relation ~ on nonzero fractional ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that (a)I = (b)J. (Here the notation (a) means the principal ideal of R consisting of all the multiples of a.) It is easily shown that this is an equivalence relation. The equivalence classes are called the ideal classes of R. Ideal classes can be multiplied: if denotes the equivalence class of the ideal I, then the multiplication = is well-defined and commutative. The principal ideals form the ideal class which serves as an identity element for this multiplication. Thus a class has an inverse if and only if there is an ideal J such that IJ is a principal ideal. In general, such a J may not exist and consequently the set of ideal classes of R may only be a monoid.
However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except R) is a product of prime ideals.
Read more about this topic: Ideal Class Group
Famous quotes containing the word definition:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth mans fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“... we all know the wags definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Its a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was mine.”
—Jane Adams (20th century)