**52**

votes

**4**answers

4k 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 ...

**21**

votes

**1**answer

4k views

### Infinite Tensor Products

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...

**6**

votes

**2**answers

691 views

### Is there one binary operation foundational for set theory?

The membership relationship "$\epsilon$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\epsilon$". Naturally, the question arises ...

**5**

votes

**0**answers

227 views

### Adjunction algebra - is there anything similar to this in abstract algebra?

I call adjunction algebra a universal algebra with one binary operation denoted as the punctuation sign (;) "semicolon" (but I will be using only one space after it, not on both sides - to avoid going ...

**72**

votes

**2**answers

7k views

### When is the tensor product of two fields a field?

Consider two extension fields $K/k, L/k$ of a field $k$.
A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by ...

**26**

votes

**9**answers

5k 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.

**22**

votes

**7**answers

2k views

### Invariant Polynomials under a Group Action (hidden GIT)

Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$).
Now the symmetric group $\mathfrak{S}_n$ ...

**29**

votes

**3**answers

3k views

### Feit-Thompson Theorem: The Odd Order Paper

For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of ...

**5**

votes

**1**answer

541 views

### Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?
...

**5**

votes

**2**answers

2k views

### Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic.

Can anyone give me an example of:
An infinite abelian but non-cyclic group whose automorphism group is cyclic.

**12**

votes

**1**answer

788 views

### Number of idempotent $n\times n$ matrices over $\mathbb{Z}_m$ ?

Is there any known formula for the number of idempotent $n\times n$ matrices over $\mathbb{Z}_m$ ?
The number of idempotent matrices over a finite field is well-known and since we can decompose $m$ ...

**3**

votes

**0**answers

253 views

### Is there any research of universal algebras axiomatized by non-Horn clauses?

A Horn clause in the language of a universal algebra is a disjunction of equations and of at most one inequality
("equation" and "inequality" are the terms used by A.Horn in his paper "On sencences ...

**2**

votes

**0**answers

276 views

### Rewrite sum of radicals equation as polynomial equation

My question is about a method described in Dr.Math forum for simplifying equations involving sums of radical functions.
(The following is a transcription of the example given by Dr. Vogler):
--- ...

**78**

votes

**18**answers

8k 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 ...

**34**

votes

**17**answers

5k 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]."
...

**75**

votes

**2**answers

3k 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 $A \not\cong A \oplus \mathbb{Z}$?

**25**

votes

**7**answers

3k 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 ...

**16**

votes

**2**answers

2k views

### Does any textbook take this approach to the isomorphism theorems?

Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______, but groups will get the points across. My ...

**21**

votes

**6**answers

2k views

### How fast are a ruler and compass?

This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO.
Consider the standard assumptions ...

**18**

votes

**3**answers

447 views

### Local Inverse Galois Problem

It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedian field $K$ has solvable (in fact supersolvable) Galois group $G$. One sees this by using the ramification filtration ...

**12**

votes

**1**answer

488 views

### How is a descent datum the same as a comodule structure?

For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong ...

**11**

votes

**2**answers

587 views

### Obstructions for a group to be the multiplicative group of a field [duplicate]

It is well known that every finite multiplicative subgroup of a field is cyclic.
I somehow got interested in a possible reverse implication:
Assume we have an abelian group $G$ whose every finite ...

**11**

votes

**3**answers

1k views

### Sign of infinite permutations?

Let $S_\infty$ the group of permutations of $\mathbb{N}$. It can be shown that there is no homomorphism $S_\infty \to \mathbf{Z}/2$ extending the sign on the finite symmetric groups. Is it possible to ...

**9**

votes

**1**answer

423 views

### Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...

**16**

votes

**2**answers

4k views

### Generalization of the shakehands/condom puzzle?

The classic handshake puzzle goes something like this:
"Given that everyone has a different skin disease, how can you safely shake hands with 3 people when you have only 2 gloves?"
Its common ...

**3**

votes

**1**answer

410 views

### How does Constructive Quantum Field Theory work?

Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via ...

**3**

votes

**1**answer

285 views

### Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$.
The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$
has the bracket
$$[xt^r, ...

**5**

votes

**2**answers

547 views

### Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $Gal(T(f))$ $\simeq$ The commutator subgroup of $Gal(f)$?

I asked this question at MSE but I did not receive an answer. So I ask it at MO:
We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by ...

**5**

votes

**0**answers

117 views

### How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...

**4**

votes

**4**answers

500 views

### What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...

**3**

votes

**1**answer

808 views

### Commutative associative rational binary operations

What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)
Feel free to re-tag if you can think of ...

**3**

votes

**2**answers

413 views

### Double orthogonal complement of a finite module

Crossposted from math.stackexchange since I'm not getting any answer.
Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the ...

**2**

votes

**1**answer

388 views

### When does $R [x]/I $ has infinitely many idempotents?

Let $R$ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotents and an unextended ideal $I$ in $R[x]$ such that $R[x]/I$ has ...

**2**

votes

**0**answers

77 views

### Polynomial constraints triggered by irreducibility [closed]

I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic:
$$af^2 + bf + c = 0$$
If we're working in a ring, ...

**5**

votes

**0**answers

217 views

### Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$

I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question
In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...

**5**

votes

**1**answer

458 views

### Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...

**4**

votes

**2**answers

570 views

### Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...

**1**

vote

**0**answers

163 views

### Separability of a simple ring extension

Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...

**1**

vote

**2**answers

432 views

### Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?

Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that ...

**0**

votes

**1**answer

129 views

### Is it possible to generalize a result of Wang?

Assume $A$ and $B$ are commutative algebras with $1$.
There is a nice result of Wang, Corollary 8, which says the following: "Let $B = A[z] = A[Z]/(h(Z))$. Then $B$ is a separable algebra over $A$ if ...

**0**

votes

**1**answer

144 views

### Maximal group image!

How to prove that if S is a finitely generated Clifford semigroup its maximal group image is actually the S_{e_{n}}?