Abstract Algebra and Algebraic Number Theory

4.3 Dedekind Domain
Definition 4.3.1. An integral domain satisfying following conditions
1. A is Noetherian ring.
2. A is integrally closed.
3. Every non zero prime ideal of A is maximal ideal.
is called a Dedekind domain
Note. Every PID satisfies the above properties and is therefore a Dedekind
domain.
Theorem 4.3.1. In
L — B
| |
K — A
setup, if A is a Dedekind domain, so is B. In particular, ring of algebraic
integer of number field is a Dedekind domain.
Definition 4.3.2 (Fractional Ideal). Let R be an integral domain with
fraction field K, let I be a R-submodule of K. I is said to be a fraction
ideal of R if rI ⊆ R for some r ∈ R ∗ . r is called denominator of factional
ideal I.
Note. An ordinary ideal of R is fractional ideal with denominator 1.
Definition 4.3.3 (Product of Ideals). Product of two ideals I and J is
the ideal generated by the product set IJ. Similarly we can define a product
of finitely many ideals.
Note. If a prime ideal P contains a product of finitely many ideals I 1 I 2 ....I n ,
then P contains I j for some j.
Proposition 4.3.2. Let R be an integral domain with fraction field K.
1. If I is finitely generated R-submodule of K, then I is a fractional ideal.
2. If R is Noetherian and I is fractional ideal of R, then I is finitely
generated R-submodule of K.
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