fractional ideal

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fractional ideal

Wikipedia

Noun

fractional ideal (plural fractional ideals)

1. (algebra, ring theory) Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R.
   • 1994, I. Martin Isaacs, Algebra: A Graduate Course, American Mathematical Society, page 476:

     Products of fractional ideals are again fractional ideals, since if ${\displaystyle A\alpha \subseteq R}$ and ${\displaystyle B\beta \subseteq R}$, then ${\displaystyle (AB)(\alpha \beta )\subseteq R}$.

   • 2001, H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, page 10:

     Theorem 5 The non-zero fractional ideals of a Dedekind domain form a multiplicative group.

   • 2008, Jan Hendrik Bruinier, Hilbert Modular Forms and Their Applications, Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, Don Zagier (editors), The 1-2-3 of Modular Forms: Lectures at a Summer School, Springer, page 106,

     A fractional ideal of ${\displaystyle F}$ is a finitely generated ${\displaystyle {\mathcal {O}}_{F}}$-submodule of ${\displaystyle F}$. Fractional ideals form a group together with the ideal multiplication. The neutral element is ${\displaystyle {\mathcal {O}}_{F}}$ and the inverse of a fractional ideal ${\displaystyle {\mathfrak {a}}\subset F}$ is

     ${\displaystyle {\mathfrak {a}}^{-1}=\left\{x\in F;x{\mathfrak {a}}\subset {\mathcal {O}}_{F}\right\}}$.

     Two fractional ideals ${\displaystyle a,b}$ are called equivalent, if there is a ${\displaystyle r\in F}$ such that ${\displaystyle a=rb}$.

Usage notes

Fractional ideals are not (generally) ideals: in some sense, fractional ideals of an integral domain are like ideals where denominators are allowed.

In particular in commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is especially useful in the study of Dedekind domains. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are for clarity sometimes termed integral ideals.

Synonyms

• (R-submodule of Frac(R) such that for some nonzero r∈R, rI ⊆ R): invertible ideal