fractional idealとは 意味・読み方・使い方

名詞

fractional ideal (複数形 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 ${\displaystyle a=rb}$.