3k views

### A characterization of invertible fractional ideals of an integral domain

Let $A$ be an integral domain, $K$ its field of fractions. Let $M$ be a fractional ideal of $A$. I'd like to prove that $M$ is invertible if and only if $MA_P$ is a principal fractional ideal of $A_P$ ...
453 views

### Necessary and sufficient condition that a localization of an integral domain is integrally closed

Is the following proposition true? Proposition? Let $A$ be an integral domain, $K$ its the field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-...
1k views

### Conductor of a quadratic order

We need some definitions to state the problem. Let $B$ be a commutative ring, $A$ its subring. We denote by $(A : B)$ the set $\{x \in B | xB \subset A\}$. $(A : B)$ is an ideal of $B$. It is ...
917 views

### The group of invertible fractional ideals of a Noetherian domain of dimension 1

Let $A$ be a Noetherian domain of dimension 1. Let $I(A)$ be the group of invertible fractional ideals of $A$. Let $P$ be a maximal ideal of $A$. Let $I(A_P)$ be the group of invertible fractional ...
432 views

### An exact sequence on the ideal class group of a Noetherian domain of dimension 1

Let $A$ be a Noetherian domain of dimension 1. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated as an $A$-module. It is well-known ...
430 views

### A lemma on the integral closure of a Noetherian domain of dimension 1

I need to prove the following lemma(?) which is motivated by this and this. Lemma Let $A$ be a Noetherian domain of dimension 1. Let $K$ be the field of fractions of $A$. Let $B$ be the integral ...
423 views

183 views

### Decomposition of a primitive ideal of a quadratic order

Let $K$ be a quadratic number field. Let $\mathcal{O}_K$ be the ring of algebraic integers in $K$. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because ...
We need some notation before we state the problem. Let $K$ be a quadratic number field, $d$ its discriminant. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of ...
### Norm of a regular ideal of an order of an algebraic number field which is Galois over $\mathbb{Q}$
Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank \$...