Definition and Basic Results
Let R be an integral domain, and let K be its field of fractions. A fractional ideal of R is an R-submodule I of K such that there exists a non-zero r ∈ R such that rI ⊆ R. The element r can be thought of as clearing out the denominators in I. The principal fractional ideals are those R-submodules of K generated by a single nonzero element of K. A fractional ideal I is contained in R if, and only if, it is an ('integral') ideal of R.
A fractional ideal I is called invertible if there is another fractional ideal J such that IJ = R (where IJ = { a1b1 + a2b2 + ... + anbn : ai ∈ I, bi ∈ J, n ∈ Z>0 } is called the product of the two fractional ideals). The set of invertible fractional ideals form an abelian group with respect to above product, where the identity is the unit ideal R itself. This group is called the group of fractional ideals of R. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an R-module.
Every finitely generated R-submodule of K is a fractional ideal and if R is noetherian these are all the fractional ideals of R.
Read more about this topic: Fractional Ideal
Famous quotes containing the words definition, basic and/or results:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animalsjust as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“It is a strange fact that freedom and equality, the two basic ideas of democracy, are to some extent contradictory. Logically considered, freedom and equality are mutually exclusive, just as society and the individual are mutually exclusive.”
—Thomas Mann (18751955)
“Being a parent is unlike any previous jobthe results of any one action are not clearly visible for a long time, if at all.”
—Anonymous Mother. As quoted in Between Generations by Ellen Galinsky, ch. 2 (1981)