All Questions
Tagged with big-list ra.rings-and-algebras
19 questions
47
votes
10
answers
6k
views
Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
1
vote
0
answers
233
views
Results that hold for the complex numbers but not for algebraically closed fields of characteristic zero
When a result is stated for the field of complex numbers it can usually be extended to a result for an algebraically closed field of characteristic zero. I would like to see a list of results that ...
- 157
71
votes
28
answers
8k
views
Results from abstract algebra which look wrong (but are true)
There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
34
votes
8
answers
4k
views
Uncountable counterexamples in algebra
In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
5
votes
0
answers
788
views
Rings such that torsion-free/flat/projective modules are flat/projective/free
While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
4
votes
1
answer
416
views
Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$
Edit: According to the comment of Todd Trimble, I revise the question.
What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...
8
votes
1
answer
716
views
Topological fraction rings and fields
Linked to this question
and as a sequel to my answer of it.
Let $R$ be a topological (commutative, unital) ring and set $S$ be a submonoid of $(R,\times,1_R)$.
Let
$$
s_{frac}\ :\ R\times S\to S^{-...
- 3,738
28
votes
9
answers
5k
views
Applications of algebra to analysis
EDIT: I would like to make a list of modern applications of algebra in analysis. By "modern" I will mean developments since the beginning of the 20th century. It is well known that classical linear ...
7
votes
1
answer
287
views
Which known theorems of Lie algebras are still valid for Leibniz algebras?
Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. Thus, it is common to see a lot of papers which topic is about a generalization of a classic theorem of Lie algebras ...
5
votes
1
answer
226
views
Examples of noncommutative Bezout domains
I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...
12
votes
7
answers
1k
views
Properties of rings that have an elegant description in terms of the associated category of modules
Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...
0
votes
1
answer
465
views
Weak algebraic structures
The following question can be thought as a sequel of this one.
Here I'm looking for a big list of example of weak algebraic structures: here weak means that the structure (i.e. operations) need not ...
- 3,349
7
votes
2
answers
830
views
Virtual algebraic calculation within proofs
It seems to me that the undergraduates I teach have particular difficulty with proofs that involve reasoning about algebraic calculations that arise only theoretically. Since I have in mind doing ...
24
votes
12
answers
3k
views
Constructions unique up to non-unique isomorphism
1) Fields have algebraic closures unique up to a non-unique isomorphism.
2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism.
3) Modules have ...
5
votes
1
answer
2k
views
Examples and importance of Embedding (and Non-Embedding) Theorems
An embedding is an injective map into a universal, simpler model object. Many embedding theorems are without obstruction, in the sense that every object which you wish to embed can be embedded. ...
62
votes
25
answers
70k
views
Linear Algebra Texts?
Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...
53
votes
7
answers
14k
views
Good lattice theory books?
A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - ...
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]."
...