All Questions
Tagged with model-theory ac.commutative-algebra
29
questions
6
votes
1
answer
274
views
The algebraic structure of a line in a (Tarski) plane
By a Tarski plane (resp. plane) I understand a mathematical structure $(X,B,\equiv)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and the 4-ary congruence relation ${\equiv}...
- 41.1k
7
votes
1
answer
256
views
Algebraic proof that the monoid ring of a torsion-free monoid is reduced
In what follows, I say that a monoid $M$ is torsion-free if the $n$-th power map is injective for all $n \geq 1$. I have a proof of the following result:
Claim: if $M$ is a torsion-free commutative ...
- 363
9
votes
0
answers
256
views
Can "$\exists\mathcal{X}(R\cong C(\mathcal{X}))$" be expressed in "large" infinitary second-order logic?
Originally asked and bountied at MSE without success:
Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ...
- 20.7k
2
votes
1
answer
86
views
An exercise in fuzzy logics built from a t-norm [closed]
Consider the following t-norm:
$$
a * b = \begin{cases}
2ab, &\quad\text{if }a, b\le1/2\\
\min\{a, b\} &\quad\text{otherwise}
\end{cases}
$$
We build from it the $\...
- 421
4
votes
1
answer
213
views
Comparing the first-order theories of different kinds of local rings of a complex variety
Let $X$ be a complex variety containing some point $x$. Then $X$ is naturally a complex-analytic space, and we have an inclusion of rings $\mathbb{C}[X]_x\hookrightarrow\mathbb{C}\{X\}_x\...
- 1,837
7
votes
0
answers
283
views
Generic behavior of "polynomialish" models of $\mathsf{Q}$
(This question was originally asked and bountied at MSE - with different notation, some more explicit arguments, and topology in place of forcing.)
Suppose $\mathcal{R}=(R_i)_{i\in\mathbb{N}}$ is a ...
- 20.7k
2
votes
0
answers
77
views
When are classes with prescribed reducts "pseudo"-elementary?
Let $\mathsf{Set}$ be the class of all sets and let $\mathcal{L}$ be a first-order language. Let $M \subseteq \mathsf{Set}$ be a set of $\mathcal{L}$-structures and let $$\mathfrak{Th}_{\in}(M) = \\ \{...
- 365
10
votes
1
answer
400
views
Is $\mathbb{Z}$ universally definable in any number fields other than $\mathbb{Q}$?
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. My question is, are there any other number fields in which $\mathbb{Z}$ is universally ...
- 4,527
6
votes
1
answer
275
views
Algebraic geometry additionally equipped with field automorphism operation
I am looking for some facts on theory, which is essentially algebraic geometry but with field automorphisms added as 'basic' operations. (Precisely, I mean universal algebraic geometry for (universal) ...
- 63
6
votes
1
answer
632
views
When are projective modules closed under highly-filtered colimits?
Let $R$ be a ring. Let $Mod(R)$ be the category of left $R$-modules, and let $Proj(R) \subseteq Mod(R)$ be the full subcategory of projective $R$-modules. Let's say that $R$-projectives are closed ...
- 62.6k
1
vote
0
answers
224
views
Ax theorem for separably closed fields
For the algebraically closed fields a theorem of Ax states that any injective polynomial map from $K^n$ to $K^n$ where $n\in \mathbb{N}$ and $K$ an algebraically closed field, is bijective.
Is there ...
- 11
4
votes
0
answers
180
views
$\mathcal{C}$-filtering of modules inherited by submodules
I'll state the question about modules, but I'm open to examples in other contexts. I am not an algebraist, so please forgive any non-conventional terminology.
DEFINITION: Let $\mathcal{C}$ be a ...
- 2,211
5
votes
0
answers
327
views
Ultrapower of a field is purely transcendental
Let $F$ be a field, $I$ a set, and $U$ an ultrafilter on $I$. Is the ultrapower $\prod_U F$ a purely transcendental field extension of $F$?
According to Chapter VII, Exercise 3.6 from Barnes, Mack "...
- 2,643
23
votes
0
answers
666
views
CH and automorphisms of ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$
Notation and motivation. Given an algebraic structure $\mathbb{M}$ of cardinality at most the continuum and with countably many operations, and a nonprincipal ultrafilter $\cal{U}$ on a countably ...
- 17.3k
21
votes
1
answer
737
views
Are $\mathbb C$ , $\mathbb C[X]$ definable in $\mathbb C[[X]]$?
Let $L$ be a first-order language and $M$ be an $L$-structure. Let $D \subseteq M^n$ . Let us say $D$ is definable in $M$ if for some finite set (possibly empty) $A=\{a_1,...,a_m\} \subseteq M$ and ...
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