In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.
fractionalideals of an integral domain are like ideals where denominators are allowed. In contexts where fractionalideals and ordinary ring ideals are...
monoid: the identity element is the fractionalideal R. For any fractionalideal I, one may define the fractionalideal I ∗ = ( R : I ) = { x ∈ K ∣ x I ⊂...
non-zero fractionalideals. The quotient of the group of non-zero fractionalideals by this subgroup is the ideal class group. Two fractionalideals I and...
fractionalideals form a group under a product. The quotient of the group of fractionalideals by the subgroup of principal ideals is then the ideal class...
colon ideal because of the notation. In the context of fractionalideals, there is a related notion of the inverse of a fractionalideal. The ideal quotient...
{\displaystyle {\mathcal {I}}_{B}} be the ideal groups of A and B, respectively (i.e., the sets of nonzero fractionalideals.) Following the technique developed...
every nonzero fractionalideal of R is irreducible in the sense that it cannot be written as a finite intersection of fractionalideals properly containing...
{g}}} . The commutator subalgebra is an ideal in g {\displaystyle {\mathfrak {g}}} , in fact the smallest ideal such that the quotient Lie algebra is abelian...
divisor. The only ideal in the zero ring is the zero ideal {0}, which is also the unit ideal, equal to the whole ring. This ideal is neither maximal...
a fractionalideal of K containing OK. By definition, the different ideal δK is the inverse fractionalideal I−1: it is an ideal of OK. The ideal norm...
groups of fractionalideals. The larger one, Im, is the group of all fractionalideals relatively prime to m (which means these fractionalideals do not...
\operatorname {Cart} (R)} is the set of fractionalideals of R. If R is a regular domain (i.e., regular at any prime ideal), then Pic(R) is precisely the divisor...
and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows: a ~ b if and only if a − b is in I. Using the ideal properties, it...
If I is an ideal of S then f−1(I) is an ideal of R. If R and S are commutative and P is a prime ideal of S then f−1(P) is a prime ideal of R. If R and...
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both...
consider right ideals and left ideals. It is common for noncommutative ring theorists to enforce a condition on one of these types of ideals while not requiring...
fractions. invertible An invertible fractionalideal is a fractionalideal that has an inverse in the monoid of fractionalideals under multiplication. irreducible...
common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring that is not...
of fractions. Then every finitely generated R-submodule I of K is a fractionalideal: that is, there is some nonzero r in R such that rI is contained in...
condition (ACC) and descending chain condition (DCC) FractionalidealIdeal class group Radical of an ideal Hilbert's Nullstellensatz Flat module Flat map Flat...
. (An ideal a {\displaystyle \mathbf {a} } as defined here is sometimes referred to as an integral ideal to distinguish from fractionalideal to be defined...
, a prime ideal p of O L {\displaystyle {\mathcal {O}}_{L}} generates the ideal pOK of O K {\displaystyle {\mathcal {O}}_{K}} . This ideal may or may...
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![{\displaystyle \mathbb {Z} [1/p]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7b4e5c999e46c7614c3264b1d80708f4363433)
