All Questions
Tagged with examples ac.commutative-algebra
34
questions
8
votes
1
answer
369
views
Noetherian but not strongly Noetherian
What are some examples of Tate rings $R$ (i.e. Huber rings with with topologically nilpotent units) which are Noetherian but not strongly Noetherian ($R$ is strongly Noetherian iff for all $n \in \...
- 1,432
11
votes
1
answer
728
views
Example of a PID with a residue field of finite characteristic and a residue field of characteristic 0?
I understand that for any nonempty set $S$ of characteristics, there exists a PID $R$ such that the set of characteristics of residue fields of $R$ (i.e. quotients by of $R$ by maximal ideals -- I'm ...
- 59k
5
votes
1
answer
140
views
Example request: seriously deficient homogeneous spaces
In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...
- 2,974
74
votes
1
answer
5k
views
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$
Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
4
votes
3
answers
850
views
Examples of (non-normal) unibranched rings?
For a local integral domain $R$ the following are equivalent:
a) The integral closure of $R$ in its fraction field (i.e., the normalization of $R$) is again local.
b) The henselization of $R$ is ...
user78294
13
votes
1
answer
584
views
Non-field example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$
I asked this question on math.stackexchange a month ago with no progress, even after a bounty. I hope to eliminate one if the other receives a satisfactory answer.
For an ideal $I\lhd R$ in a ...
- 1,593
4
votes
1
answer
445
views
Uncountable Reduced ring $R$ with $R[x]$ has only a countable number of maximal left ideals
The question is following:
Is there an uncountable reduced ring (i.e., a ring with no non-zero nilpotent element) $R$ (with identity) such that
$R[x]$ has only a countable number of maximal left ...
user39982
14
votes
1
answer
1k
views
Examples of polynomial rings $A[x]$ with relatively large Krull dimension
If $A$ is a commutative ring we have the estimate
$$
\dim (A)+1 \le \dim (A[x])\le 2\dim (A)+1
$$
for the Krull dimension, with $\dim (A)+1 = \dim (A[x])$ for Noetherian rings.
I am looking for nice ...
- 11.9k
1
vote
0
answers
168
views
Algebraic properties of the semiring of open subsets.
Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at all?...
- 3,435
2
votes
2
answers
1k
views
Infinite domain with finite number of prime ideals(elements)
While trying to prove one property of commutative rings with units I can't prove one fact without assuming existence of infinitely many different prime ideals or elements. I tried to test if it was ...
- 616
7
votes
2
answers
1k
views
The rank of a not necessarily finitely generated module.
This question is motivated by this one. The main point of the question (was) to try to weaken the notion of rank. After the answers and comments, it seems this is not a good way to do it, but perhaps ...
- 42.1k
6
votes
1
answer
2k
views
An example of a rank one projective R-Module that is not invertible
Let $R$ be a commutative noetherian ring. I know that an $R$-module is invertible iff it is finitely generated and locally free of rank one. I presume then that there are examples of non-finitely ...
- 325
12
votes
2
answers
998
views
Cohen-Macaulay domain with non-Cohen-Macaulay normalization?
Is the normalization of a Cohen-Macaulay domain necessarily Cohen-Macaulay? I suspect that the answer is no, but I don't have a counterexample. I am most interested in "geometric" situations, so one ...
- 9,982
2
votes
3
answers
1k
views
Algebraic structures of greater cardinality than the continuum?
Are there interesting algebraic structures whose cardinality is greater than the continuum? Obviously, you could just build a product group of $\beth_2$ many groups of whole numbers to get to such a ...
- 167
25
votes
5
answers
2k
views
Exotic principal ideal domains
Recently I realized that the only PIDs I know how to write down that aren't fields are $\mathbb{Z}, F[x]$ for $F$ a field, integral closures of these in finite extensions of their fraction fields that ...
13
votes
5
answers
3k
views
Example of a projective module which is not a direct sum of f.g. submodules?
This semester I am teaching a graduate course in commutative algebra, and I have been taking the occasion to try to look at the proofs of some the results in my basic source material (Matsumura, ...
- 64.4k
19
votes
6
answers
2k
views
Nonfree projective module over a regular UFD?
What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least ...
- 64.4k
28
votes
3
answers
6k
views
Non finitely-generated subalgebra of a finitely-generated algebra
Ok, I feel a little bit ashamed by my question.
This afternoon in the train, I looked for a counter-example:
— $k$ a field
— $A$ a finitely generated $k$-algebra
— $B$ a $k$-subalgebra of $A$ that ...
43
votes
5
answers
3k
views
Explicit elements of $K((x))((y)) \setminus K((x,y))$
In an answer to the popular question on common false beliefs in mathematics
Examples of common false beliefs in mathematics
I mentioned that many people conflate the two different kinds of formal ...
- 64.4k
5
votes
2
answers
1k
views
Gaining intuition for how submodules behave
I'm studying elementary commutative algebra this semester, largely following Atiyah-MacDonald. I often find myself in a situation where I'm interested in whether some property of an R-module M is ...
- 2,750
9
votes
0
answers
272
views
Krull rings and determinantal invariants
During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
- 21.8k
2
votes
1
answer
1k
views
Example of restriction of a finite morphism which is not finite
Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties $X\...
- 422
1
vote
1
answer
946
views
Example of inclusion which is not a finite morphism [closed]
Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective?
Thanks!
...
- 422
4
votes
3
answers
1k
views
An example where GCD depends on the domain
First some notation. Given a domain $R$ and $x,a,b \in R$, I write $x=gcd(a,b)_R$ to mean that $x$ is one gcd of $a$ and $b$ in $R$.
I want to find an example of an GCD-domain $R$, a subdomain $S \...
2
votes
0
answers
252
views
Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
4
votes
1
answer
1k
views
On using field extensions to prove the impossiblity of a straightedge and compass construction
Let $z \in \mathbb{C}$. Consider the following statements:
The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$.
There is a field extension $K / \mathbb{...
37
votes
2
answers
10k
views
A finitely generated, locally free module over a domain which is not projective?
This is a followup to a previous question
What is the right definition of the Picard group of a commutative ring?
where I was worried about the distinction between invertible modules and rank one ...
- 64.4k
11
votes
5
answers
7k
views
An example of two elements without a greatest common divisor
Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously.
"Easy" means that I can explain it to my ...
81
votes
30
answers
67k
views
Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
36
votes
17
answers
6k
views
Canonical examples of algebraic structures
Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.
"When I read about a [insert structure here], I immediately think of [example]."
...
8
votes
2
answers
597
views
Which commutative rigs arise from a distributive category?
A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an ...
- 12.5k
18
votes
9
answers
2k
views
What representative examples of modules should I keep in mind?
So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only ...
- 114k
32
votes
2
answers
3k
views
Graded local rings versus local rings
A lot of times I see theorems stated for local rings, but usually they are also true for "graded local rings", i.e., graded rings with a unique homogeneous maximal ideal (like the polynomial ring). ...
- 9,982
34
votes
4
answers
11k
views
Finite extension of fields with no primitive element
What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
- 23.6k