fractional ideal

1 Basics

Let $A$ be an integral domain with field of fractions $K$ . Then $K$ isan $A$ –module, and we define a fractional ideal of $A$ to be asubmodule of $K$ which is finitely generated as an $A$ –module.

The product of two fractional ideals ${\\mathfrak{a}}$ and ${\\mathfrak{b}}$ of $A$ is definedto be the submodule of $K$ generated by all the products $x\\cdot y\\in K$ , for $x\\in{\\mathfrak{a}}$ and $y\\in{\\mathfrak{b}}$ . This product is denoted ${\\mathfrak{a}}\\cdot{\\mathfrak{b}}$ , and it is always a fractional ideal of $A$ as well. Notethat, if $A$ itself is considered as a fractional ideal of $A$ , then ${\\mathfrak{a}}\\cdot A={\\mathfrak{a}}$ . Accordingly, the set of fractional ideals is alwaysa monoid under this product operation, with identity element $A$ .

We say that a fractional ideal ${\\mathfrak{a}}$ is invertible if thereexists a fractional ideal ${\\mathfrak{a}}^{\\prime}$ such that ${\\mathfrak{a}}\\cdot{\\mathfrak{a}}^{\\prime}=A$ . It canbe shown that if ${\\mathfrak{a}}$ is invertible, then its inverse must be ${\\mathfrak{a}}^{\\prime}=(A:{\\mathfrak{a}})$ , the annihilator¹¹In general, for any fractionalideals ${\\mathfrak{a}}$ and ${\\mathfrak{b}}$ , the annihilator of ${\\mathfrak{b}}$ in ${\\mathfrak{a}}$ is thefractional ideal $({\\mathfrak{a}}:{\\mathfrak{b}})$ consisting of all $x\\in K$ such that $x\\cdot{\\mathfrak{b}}\\subset{\\mathfrak{a}}$ . of ${\\mathfrak{a}}$ in $A$ .

2 Fractional ideals in Dedekind domains

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

The unique factorization of ideals theorem states that everyfractional ideal in $A$ factors uniquely into a finite product ofprime ideals of $A$ and their (fractional ideal) inverses. It followsthat the ideal group of $A$ is freely generated as an abelian group bythe nonzero prime ideals of $A$ .

A fractional ideal of $A$ is said to be principal if it isgenerated as an $A$ –module by a single element. The set of nonzeroprincipal fractional ideals is a subgroup of the ideal group of $A$ ,and the quotient group of the ideal group of $A$ by the subgroup ofprincipal fractional ideals is nothing other than the ideal classgroup of $A$ .

单词	FractionalIdeal
释义	fractional ideal 1 Basics Let $A$ be an integral domain with field of fractions $K$ . Then $K$ isan $A$ –module, and we define a fractional ideal of $A$ to be asubmodule of $K$ which is finitely generated as an $A$ –module. The product of two fractional ideals ${\\mathfrak{a}}$ and ${\\mathfrak{b}}$ of $A$ is definedto be the submodule of $K$ generated by all the products $x\\cdot y\\in K$ , for $x\\in{\\mathfrak{a}}$ and $y\\in{\\mathfrak{b}}$ . This product is denoted ${\\mathfrak{a}}\\cdot{\\mathfrak{b}}$ , and it is always a fractional ideal of $A$ as well. Notethat, if $A$ itself is considered as a fractional ideal of $A$ , then ${\\mathfrak{a}}\\cdot A={\\mathfrak{a}}$ . Accordingly, the set of fractional ideals is alwaysa monoid under this product operation, with identity element $A$ . We say that a fractional ideal ${\\mathfrak{a}}$ is invertible if thereexists a fractional ideal ${\\mathfrak{a}}^{\\prime}$ such that ${\\mathfrak{a}}\\cdot{\\mathfrak{a}}^{\\prime}=A$ . It canbe shown that if ${\\mathfrak{a}}$ is invertible, then its inverse must be ${\\mathfrak{a}}^{\\prime}=(A:{\\mathfrak{a}})$ , the annihilator¹¹In general, for any fractionalideals ${\\mathfrak{a}}$ and ${\\mathfrak{b}}$ , the annihilator of ${\\mathfrak{b}}$ in ${\\mathfrak{a}}$ is thefractional ideal $({\\mathfrak{a}}:{\\mathfrak{b}})$ consisting of all $x\\in K$ such that $x\\cdot{\\mathfrak{b}}\\subset{\\mathfrak{a}}$ . of ${\\mathfrak{a}}$ in $A$ . 2 Fractional ideals in Dedekind domains We now suppose that $A$ is a Dedekind domain. In this case, everynonzero fractional ideal is invertible, and consequently the nonzerofractional ideals in $A$ form a group under ideal multiplication,called the ideal group of $A$ . The unique factorization of ideals theorem states that everyfractional ideal in $A$ factors uniquely into a finite product ofprime ideals of $A$ and their (fractional ideal) inverses. It followsthat the ideal group of $A$ is freely generated as an abelian group bythe nonzero prime ideals of $A$ . A fractional ideal of $A$ is said to be principal if it isgenerated as an $A$ –module by a single element. The set of nonzeroprincipal fractional ideals is a subgroup of the ideal group of $A$ ,and the quotient group of the ideal group of $A$ by the subgroup ofprincipal fractional ideals is nothing other than the ideal classgroup of $A$ .
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