Fractional Ideals: Definitions, Operations, and Invertibility

Introduction

In algebraic number theory, ideals play a fundamental role in understanding the structure of rings of integers of number fields. A fractional ideal is a subset of the field of fractions of a ring that is integral over the ring. Unlike ordinary ideals, fractional ideals are not necessarily contained within the ring itself but are instead subsets of the field of fractions that satisfy specific conditions. These ideals are central to the study of algebraic integers, Dedekind domains, and the theory of divisors. The concept of fractional ideals extends the classical theory of ideals by allowing for a more flexible framework in which ideals can be non-prime yet still invertible. This article explores the definitions, operations, and properties of fractional ideals, emphasizing their significance in algebraic number theory.

Definitions

A fractional ideal of a ring $ R $ is a finitely generated $ R $-module $ N $ that is also a subring of the field of fractions $ Q(R) $. Specifically, $ N $ must satisfy the following properties:

  1. $ N $ is a finitely generated $ R $-module.
  2. $ N $ is contained in $ Q(R) $, the field of fractions of $ R $.
  3. For every nonzero element $ a \in R $, the element $ a^{-1} \in Q(R) $ is in $ N $ if and only if $ a \in N $.

This condition ensures that fractional ideals are invertible in the sense that they can be multiplied by elements of $ R $ to produce units. In contrast, ordinary ideals are subsets of $ R $, and their inverses are not necessarily in $ R $. Fractional ideals thus provide a bridge between algebraic structures and the theory of fields, enabling the study of divisibility and factorization in non-integer domains.

Operations on Fractional Ideals

The operations of addition, multiplication, and duals are central to the study of fractional ideals.

Addition

The addition of two fractional ideals $ N_1 $ and $ N_2 $ is defined as the set $ N_1 + N_2 = { a + b \mid a \in N_1, b \in N_2 } $. This operation is compatible with the ring structure of $ R $, ensuring that the resulting set is also a fractional ideal. For example, if $ N_1 $ and $ N_2 $ are fractional ideals of $ R $, then $ N_1 + N_2 $ is a fractional ideal of $ R $, and its elements are closed under the ring operations.

Multiplication

Multiplication of two fractional ideals $ N_1 $ and $ N_2 $ is defined as the set $ N_1 \cdot N_2 = { a \cdot b \mid a \in N_1, b \in N_2 } $. This operation is associative and distributive, making it analogous to the multiplication of ideals in a ring. Importantly, the product of two fractional ideals is again a fractional ideal, and the resulting ideal is generated by the products of elements from $ N_1 $ and $ N_2 $.

Duals

The dual of a fractional ideal $ N $ is defined as the set $ N^* = { a \in Q(R) \mid a \in N } $. This dual operation is particularly significant in the context of Dedekind domains, where the dual of a fractional ideal corresponds to an ideal in the ring of integers. The dual of a fractional ideal is also a fractional ideal, and the product of a fractional ideal with its dual is a unit in the ring $ R $.

Invertibility of Fractional Ideals

A fractional ideal is said to be invertible if there exists another fractional ideal $ M $ such that $ N \cdot M = R $. This property is closely tied to the structure of the ring $ R $ and the nature of its ideals. In a Dedekind domain, every fractional ideal is invertible, as the ring is a UFD and every ideal is principal. However, in more general rings, invertibility is conditional.

In a Dedekind domain, the invertibility of a fractional ideal is equivalent to the ideal being generated by a single element. This is because Dedekind domains have the property that every ideal is invertible, and the dual of a fractional ideal corresponds to an ideal in the ring of integers. In contrast, in a general ring, invertibility is only guaranteed for fractional ideals that are generated by a single element or for which their duals are invertible.

Applications and Significance

Fractional ideals are pivotal in the study of algebraic number theory, particularly in the analysis of rings of integers and their properties. In the context of number fields, fractional ideals correspond to divisors of the ring of integers, and their behavior is closely tied to the theory of ideals in number fields. The invertibility of fractional ideals in Dedekind domains ensures that the structure of these rings is highly divisible, allowing for the study of factorization and the behavior of ideals under various operations.

Moreover, fractional ideals play a crucial role in algebraic geometry, where they are used to study the structure of schemes and the behavior of divisors. In cryptography, the properties of fractional ideals are leveraged in the construction of secure algorithms, particularly in the context of lattices and ideal classes.