# fractional ideal

## 1 Basics

Let $A$ be an integral domain^{} with field of fractions^{} $K$. Then $K$ is
an $A$βmodule, and we define a fractional ideal^{} of $A$ to be a
submodule of $K$ which is finitely generated^{} as an $A$βmodule.

The product of two fractional ideals $\mathrm{\pi \x9d\x94\x9e}$ and $\mathrm{\pi \x9d\x94\x9f}$ of $A$ is defined
to be the submodule of $K$ generated by all the products $x\beta \x8b\x85y\beta \x88\x88K$, for $x\beta \x88\x88\mathrm{\pi \x9d\x94\x9e}$ and $y\beta \x88\x88\mathrm{\pi \x9d\x94\x9f}$. This product is denoted $\mathrm{\pi \x9d\x94\x9e}\beta \x8b\x85\mathrm{\pi \x9d\x94\x9f}$, and it is always a fractional ideal of $A$ as well. Note
that, if $A$ itself is considered as a fractional ideal of $A$, then
$\mathrm{\pi \x9d\x94\x9e}\beta \x8b\x85A=\mathrm{\pi \x9d\x94\x9e}$. Accordingly, the set of fractional ideals is always
a monoid under this product operation, with identity element^{} $A$.

We say that a fractional ideal $\mathrm{\pi \x9d\x94\x9e}$ is invertible if there
exists a fractional ideal ${\mathrm{\pi \x9d\x94\x9e}}^{\beta \x80\xb2}$ such that $\mathrm{\pi \x9d\x94\x9e}\beta \x8b\x85{\mathrm{\pi \x9d\x94\x9e}}^{\beta \x80\xb2}=A$. It can
be shown that if $\mathrm{\pi \x9d\x94\x9e}$ is invertible, then its inverse^{} must be ${\mathrm{\pi \x9d\x94\x9e}}^{\beta \x80\xb2}=(A:\mathrm{\pi \x9d\x94\x9e})$, the annihilator^{}^{1}^{1}In general, for any fractional
ideals $\mathrm{\pi \x9d\x94\x9e}$ and $\mathrm{\pi \x9d\x94\x9f}$, the annihilator of $\mathrm{\pi \x9d\x94\x9f}$ in $\mathrm{\pi \x9d\x94\x9e}$ is the
fractional ideal $(\mathrm{\pi \x9d\x94\x9e}:\mathrm{\pi \x9d\x94\x9f})$ consisting of all $x\beta \x88\x88K$ such that
$x\beta \x8b\x85\mathrm{\pi \x9d\x94\x9f}\beta \x8a\x82\mathrm{\pi \x9d\x94\x9e}$. of $\mathrm{\pi \x9d\x94\x9e}$ in $A$.

## 2 Fractional ideals in Dedekind domains

We now suppose that $A$ is a Dedekind domain^{}. In this case, every
nonzero fractional ideal is invertible, and consequently the nonzero
fractional ideals in $A$ form a group under ideal multiplication,
called the ideal group of $A$.

The unique factorization^{} of ideals theorem states that every
fractional ideal in $A$ factors uniquely into a finite product of
prime ideals^{} of $A$ and their (fractional ideal) inverses. It follows
that the ideal group of $A$ is freely generated as an abelian group^{} by
the nonzero prime ideals of $A$.

A fractional ideal of $A$ is said to be principal if it is
generated as an $A$βmodule by a single element. The set of nonzero
principal fractional ideals is a subgroup^{} of the ideal group of $A$,
and the quotient group^{} of the ideal group of $A$ by the subgroup of
principal fractional ideals is nothing other than the ideal class
group^{} of $A$.

Title | fractional ideal |
---|---|

Canonical name | FractionalIdeal |

Date of creation | 2013-03-22 12:42:38 |

Last modified on | 2013-03-22 12:42:38 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 5 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 13A15 |

Classification | msc 13F05 |

Related topic | IdealClassGroup |

Defines | ideal group |