2
votes
1answer
108 views

Factor ring by a regular ideal of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is a finitely generated $A$-module. It is well-known that ...
5
votes
1answer
214 views

Using Nakayama's Lemma to prove isomorphism theorem for finitely generated free modules

Suppose $R \neq 0$ is a commutative ring with $1$. The following is well known: (Isomorphism Theorem for Finitely Generated Free Modules) [FGFM] $R^{n}\cong R^{m}$ as $R$-modules if and only if ...
6
votes
1answer
232 views

The ring of integers of a number field is finitely generated.

For a number field $K$, we define the ring of integers of $K$ to be $$\mathcal{O}_K:=\{x\in K\big|\ (\exists f\in\mathbb{Z}[X])(f\ \text{ is monic and } f(x)=0)\}.$$ Is there any easy way to see from ...
5
votes
1answer
163 views

Tensor-free proof that for finite modules over reduced Noetherian rings, locally free = projective

Is there an elegant tensor-free proof of the fact that over a reduced Noetherian ring $A$, every finitely-generated $A$-module which is locally free, is projective? EDIT: I would be content with the ...