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2 votes
2 answers
416 views

Tensor product over $\mathbb{Z}$ and p-adic integer ring $\mathbb{Z}_p$

Thanks for your reading. Suppose we have two $\mathbb{Z}_p$-modules $A,B$. Do we always have $A \otimes_{\mathbb{Z}} B \simeq A \otimes_{\mathbb{Z}_p} B$, as abelian groups or $\mathbb{Z}_p$-modules? ...
Rellw's user avatar
  • 319
5 votes
0 answers
181 views

The precise relationship between (moduli space of) finite-dimensional commutative local $\kappa$-algebras and number theory?

In [BP08] Poonen constructs and studies $\mathfrak{B}_n$, the moduli space of all based $n$-dimensional commutative associative unital $\kappa$-algebras, where $\kappa$ is an algebraically closed ...
M.G.'s user avatar
  • 7,127
5 votes
1 answer
335 views

About the structure of unit groups appearing in number theory

I think the following statement is not true in the general situations, but consider it: $R$ is a ring, $\mathfrak{p}$ is a prime ideal, then the unit group of $\dfrac{R}{\mathfrak{p}^nR}$ is ...
Tireless and hardworking's user avatar
3 votes
0 answers
79 views

The type number of an algebra

I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \...
a196884's user avatar
  • 323
2 votes
1 answer
216 views

A problem about an unramified prime in a Galois extension

I asked this question in MathStackExchange, but I didn't receive any answer. Let $K/\mathbb{Q}$ be a Galois extension of degree $n$, and denote its ring of integers by $\mathcal{O}_K$. Let $\mathfrak{...
Tireless and hardworking's user avatar
2 votes
0 answers
147 views

Characterization of algebraic integers providing a prime ideal

Let $\alpha$ be an algebraic integer and let $\mathcal{O}_{\mathbb{Q}(\alpha)}$ be the ring of integers of $\mathbb{Q}(\alpha)$. Question: How to characterize an algebraic integer $\alpha$ such that $\...
Sebastien Palcoux's user avatar
1 vote
1 answer
142 views

Valuation theory on semisimple algebras used in the paper of Cohen-Martinet: reference request

I'm currently reading the paper of Henri Cohen & Jacques Martinet "Etude heuristique des groupes de classes des corps de nombres" On the 2nd section, they recall some facts on valuations, ...
gualterio's user avatar
  • 1,013
2 votes
1 answer
746 views

Motivation to study the order theory (ring theory)

I'm currently reading a paper of Georges Gras on the Reflection Principle. The paper uses some theorems about orders (ring theory) from the book "Maximal Orders" by Reiner. I find the book interesting,...
gualterio's user avatar
  • 1,013
1 vote
2 answers
685 views

Every ideal is principal in a certain quotient ring

Does anybody know if there is a valuation-free proof of the following fact? Let $K$ be a finite extension of $\mathbb{Q}$. Then, for every $q\in \mathbb{Z}$, the quotient ring $\mathcal{O}_{K}/\...
Jamai-Con's user avatar
4 votes
1 answer
173 views

On the factorization of powers of atoms in the ring of integers of a number field

Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is a non-unit element $a \in H$ that doesn't split into the product of two non-unit elements. Given $x \in H$, we ...
Salvo Tringali's user avatar
1 vote
0 answers
50 views

Closedness of the range of the distorsion of the multiplicative monoid of a number field

Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is an element $x \in H \setminus H^\times$ such that $a \ne xy$ for all $x, y \in H \setminus H^\times$, where $H^\...
Salvo Tringali's user avatar
2 votes
1 answer
216 views

Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:SL_n(R)\rightarrow SL_n(R/I) $$ (In the original question I had put $GL_n$ instead of $SL_n$ ...
Hugo Chapdelaine's user avatar
1 vote
0 answers
189 views

What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [closed]

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
eddie's user avatar
  • 255