All Questions
Tagged with rings-and-algebras or ra.rings-and-algebras
3,500 questions
123
votes
18
answers
14k
views
How do you decide whether a question in abstract algebra is worth studying?
Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...
114
votes
2
answers
12k
views
How would you solve this tantalizing Halmos problem?
$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric ...
- 4,736
96
votes
16
answers
18k
views
Why is it a good idea to study a ring by studying its modules?
This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...
- 118k
95
votes
11
answers
6k
views
Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?
Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
- 2,600
94
votes
2
answers
7k
views
$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$
Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
- 63.1k
87
votes
5
answers
10k
views
When is $A$ isomorphic to $A^3$?
This is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a ...
- 63.1k
76
votes
9
answers
6k
views
Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
The question is the extent to which we can unify addition
and multiplication, realizing them as terms in a single
underlying binary operation. I have a number of questions.
Is there a binary ...
- 236k
74
votes
1
answer
6k
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 ...
- 63.1k
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 ...
64
votes
4
answers
8k
views
What is the current status of the Kaplansky zero-divisor conjecture for group rings?
Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good ...
- 1,065
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 ...
62
votes
5
answers
10k
views
Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can ...
- 11.2k
61
votes
3
answers
7k
views
Why is there no Cayley's Theorem for rings?
Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...
- 3,230
56
votes
2
answers
5k
views
A condition that implies commutativity
Let $R$ be a ring. A notable theorem of N. Jacobson states that if the identity $x^{n}=x$ holds for every $x \in R$ and a fixed $n \geq 2$ then $R$ is a commutative ring.
The proof of the result for ...
- 8,842
55
votes
0
answers
2k
views
What did Gelfand mean by suggesting to study "Heredity Principle" structures instead of categories?
Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the ...
- 1,238
53
votes
5
answers
5k
views
Does this formula have a rigorous meaning, or is it merely formal?
I hope this problem is not considered too "elementary" for MO. It concerns a formula that I have always found fascinating. For, at first glance, it appears completely "obvious", while on closer ...
- 15.3k
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 - ...
52
votes
7
answers
8k
views
"Algebraic" topologies like the Zariski topology?
The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...
51
votes
1
answer
2k
views
Invertible matrices over noncommutative rings
Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?
The question popped up ...
- 2,941
47
votes
9
answers
11k
views
What are the reasons for considering rings without identity?
I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.
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 ...
47
votes
2
answers
3k
views
Non isomorphic finite rings with isomorphic additive and multiplicative structure
About a year ago, a colleague asked me the following question:
Suppose $(R,+,\cdot)$ and $(S,\oplus,\odot)$ are two rings such that $(R,+)$ is isomorphic, as an abelian group, to $(S,\oplus)$, and $...
- 7,207
45
votes
8
answers
6k
views
What makes a theorem *a* "nullstellensatz."
I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...
44
votes
2
answers
2k
views
Fermat's Last Theorem for integer matrices
Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
- 1,889
43
votes
18
answers
5k
views
Results in linear algebra that depend on the choice of field
Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers).
I am looking for a list of concepts, and results, in ...
43
votes
3
answers
7k
views
transcendental Galois theory
Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ of $K \setminus L$, ...
- 65.4k
42
votes
6
answers
7k
views
An algebra of "integrals"
When discussing divergent integrals with people, I got curious about the following:
Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace)
$$\int_0^{\infty}: ...
- 3,806
41
votes
7
answers
6k
views
Why don't ideals and quotients work well for categories?
Ideals are intimately related to quotients and congruence relations. They clearly play a very important role in ring theory and order theory. So do normal subgroups in group theory. (Enriched) ...
- 3,937
41
votes
4
answers
33k
views
What is the intuition for the trace norm (nuclear norm)?
I will word this question in terms of linear operators acting on $\mathbb{C}^n$ for simplicity. Feel free to provide an answer in terms of more general Hilbert spaces if you think it makes more sense ...
- 529
40
votes
9
answers
10k
views
Simplest examples of rings that are not isomorphic to their opposites
What are the simplest examples of
rings that are not isomorphic to their
opposite rings? Is there a science to constructing them?
The only simple example known to me:
In Jacobson's Basic Algebra (...
- 5,654
39
votes
5
answers
5k
views
When does a ring surjection imply a surjection of the group of units?
The following might be a very trivial question. If so, I don't mind it being closed, but would appreciate a reference where I could read about it.
Let $R$ and $S$ be commutative rings and let $R^\...
- 33.3k
39
votes
4
answers
5k
views
Is there a universal property for Witt vectors?
Do the Witt vectors satisfy a universal property?
- 7,994
39
votes
5
answers
4k
views
Is there an explicit construction of a free coalgebra?
I am interested in the differences between algebras and coalgebras. Naively, it does not seem as though there is much difference: after all, all you have done is to reverse the arrows in the ...
- 9,132
38
votes
3
answers
3k
views
What is the current status of Agrawal's conjecture?
In their famous 'Primes is in P' paper Agrawal, Kayal and Saxena stated the following conjecture:
If for coprime integers $n$ and $r$ the equality $(X-1)^n = X^n - 1$ holds in $\mathbb{Z}_n[X]/(X^r-...
- 463
38
votes
0
answers
1k
views
Groups whose complex irreducible representations are finite dimensional
By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...
- 38.6k
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]."
...
36
votes
1
answer
3k
views
Whence “homomorphism” and “homomorphic”?
Today homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen?
“Homomorphic” (and “homomorphism” as “property of being ...
- 31.5k
36
votes
3
answers
2k
views
Are large powers of polynomials linearly independent?
Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
- 4,653
36
votes
4
answers
5k
views
What is interesting/useful about big Witt Vectors?
$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\...
- 10.5k
35
votes
3
answers
3k
views
When is a power of a nonnegative polynomial a sum of squares?
There are nonnegative polynomials that are not sums of squares. For example Motzkin gave the example $x^4y^2+x^2y^4+z^6-3x^2y^2z^2$ in 1967.
Is there a real polynomial $f\in{\mathbb{R}}[x_1,\dotsc,x_n]...
user2529
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 ...
34
votes
1
answer
5k
views
Freyd-Mitchell's embedding theorem
Freyd–Mitchell's embedding theorem states that: if $A$ is a small abelian category, then there exists a ring R and a full, faithful and exact functor $F\colon A \to R\text{-}\mathrm{Mod}$.
I have been ...
- 3,004
33
votes
2
answers
7k
views
Dimension of infinite product of vector spaces
This question is motivated by the question link text, which compares the infinite direct sum and the infinite direct product of a ring.
It is well-known that an infinite dimensional vector space is ...
- 20.8k
33
votes
2
answers
2k
views
What do cluster algebras tell us about Grassmannians?
One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
- 1,690
32
votes
7
answers
6k
views
Consequences of not requiring ring homomorphisms to be unital?
As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example ...
- 6,782
32
votes
3
answers
4k
views
Example for column rank $\neq$ row rank
The proof that column rank = row rank for matrices over a field relies on the fact that the elements of a field commute. I'm looking for an easy example of a matrix over a ring for which column rank $\...
- 4,014
32
votes
1
answer
17k
views
The gimbal lock shows up in my quaternions
I suspect this is a bit basic for mathoverflow, seeing I'm still just an undergraduate
I've been playing around with quaternions as means to eliminate the gimbal lock. From what I understand, one ...
32
votes
7
answers
4k
views
"Sums-compact" objects = f.g. objects in categories of modules?
Hello,
Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. ...
- 5,562
32
votes
1
answer
3k
views
Have you ever seen this bizarre commutative algebra?
I have encountered very strange commutative nonassociative algebras without unit, over a characteristic zero field, and I cannot figure out where do they belong. Has anybody seen these animals in any ...
- 18.4k
32
votes
1
answer
2k
views
Is this formal noncommutative power series identity known?
I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series
$$ 1 + \...
- 114k