The abstract-algebra tag has no wiki summary.

**0**

votes

**3**answers

1k views

### terminology about ring/algebra in abstract algebra and measure theory

Both in abstract algebra and measure theory is there term ring/algebra, but their definition are different and we can not deduce one from the other, the only requirement in definition they share is ...

**22**

votes

**9**answers

4k 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.

**9**

votes

**2**answers

784 views

### Free division rings?

Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators?

**1**

vote

**1**answer

457 views

### Is the multiplication beetween even numbers an associative algebra?

We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist?
It has been proposed as a counterexample the set of even numbers. ...

**4**

votes

**1**answer

350 views

### Mechanically instantiating abstract constructions

I am looking for work on the effective inverse of abstraction, aka specialization.
There are two ways in which abstraction helps us:
Get a better understanding of the structural rules at play in ...

**12**

votes

**5**answers

1k views

### Are rings really more fundamental objects than semi-rings?

The discovery (or invention) of negatives, which happened several centuries ago by the Chinese, Indians and Arabs, has of course be of fundamental importance to mathematics.
From then on, it seems ...

**1**

vote

**1**answer

200 views

### How to distinguish property of particular representation from property of algebraic structure?

It is common that You have some interesting object (set, group, algebra or something, whatever) which has certain properties, structure etc. You may try describe it in pure algebraic way. Sometimes ...

**6**

votes

**1**answer

2k views

### On order of subgroups in abelian groups

I wonder whether any of you guys has already read the homonymous note by R. Beals in the December 2009 issue of the Monthly.
If so, would you be so kind as to let me know about the main ideas in ...

**6**

votes

**8**answers

4k views

### Best way to teach concept of real numbers using a hands-on activity?

I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.

**3**

votes

**1**answer

279 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, ...

**2**

votes

**2**answers

862 views

### about state-field correspondence

In the defination of vertex algebra,we call the vertex operator state-field correspondence,does that mean that it is an injective map??
Are there some physical intepretations about state-field ...

**4**

votes

**2**answers

2k views

### Using Weierstrass’s Factorization Theorem

I am trying to factorize $\sin(x)\over x$ which by Taylor series expansion and using the roots is $$a \cdot \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} ...

**2**

votes

**1**answer

162 views

### Introduction to modular property of affine alegebra and conformal vertex algebra

I wonder how modular property naturally arises in conformal theory.
Is it obvious from physical viewpoint?

**0**

votes

**1**answer

211 views

### About vertex algebra ,mode expansion

A vertex operator is a linear map associating every state to a operator-valued distributions(quantum field) on a algebra curve,which is also called operator-state correspondence.
Chose a local complex ...

**7**

votes

**4**answers

2k views

### Variants of Eisenstein irreducibility

In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he ...

**2**

votes

**1**answer

911 views

### Unique factorization in polynomial rings

Everybody knows that polynomial rings over fields have unique factorization, and that if $R$ has unique factorization, then so does $R[X]$. And everybody knows who proved these results first.
Well, ...

**9**

votes

**2**answers

1k views

### Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.

**0**

votes

**2**answers

237 views

### Existence of an “anti-additive” (or “never linear”) map?

(I've edited this question)
I'm searching for a continuously differentiable function $f:\mathbb R^2\to\mathbb R$ such that $f(x)+f(x+u+v)\neq f(x+u)+f(x+v)$ for all $x$ and all linearly independent ...

**17**

votes

**4**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 $\mathfrak{S}_n$ ...

**3**

votes

**2**answers

661 views

### Injective modules and Pontrjagin duals

Forgive me for this naive question.
We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, page 784.
Every module is a submodule of an ...

**4**

votes

**2**answers

721 views

### Is \mathbb{C}^{n \times n} algebraically closed?

Inspite of the fact that $C^{n \times n}$ is not a field, is it still possible to talk about it being 'algebraically closed' in the sense that $\forall f \in \mathbb{C}^{n \times n}[x]$ does $\exists ...

**6**

votes

**2**answers

524 views

### Examples of Completions and Algebraic Closures

It is widely known that the algebaric closure of the $p$-adic completion $\mathbb{Q}_p$ of $\mathbb{Q}$ isn't complete anymore. It's completion is complete and known as $\mathbb{C}_p$.
I have read ...

**11**

votes

**3**answers

999 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 ...

**20**

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 ...

**15**

votes

**4**answers

7k views

### Famous exercise from Lang's Algebra

There's a famous story about an exercise from Lang's Algebra that says something along the lines of "pick up a homological algebra book and prove all of the theorems yourself". I cannot find it in ...

**4**

votes

**2**answers

831 views

### when are epimorphisms of algebraic objects surjective?

let $C$ be the category of $\tau$-algebras for some type $\tau$. consider the statements:
every monomorphism is regular.
every epimorphism in C is surjective.
it is easy to see that 1. implies 2. ...

**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 ...

**42**

votes

**4**answers

3k 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 ...

**0**

votes

**1**answer

3k views

### Newbie boolean algebra question [closed]

The Majority Function is
1: A.B.¬C + A.¬B.C + ¬A.B.C + A.B.C
I can see intuitively that it can be simplified to
2: A.B + A.C + B.C
and thus A.(B + C) + B.C
but how can I use boolean algebra to ...

**3**

votes

**2**answers

771 views

### computation, algebra, logic

So a really simple way of describing a digital computer is to say that it is a device for performing boolean operations. You feed it a bunch of bit strings, which is a description of the problem and ...

**2**

votes

**2**answers

181 views

### Decomposition result for multivariate polynomial

Let $k$ be a positive integer greater than $1$ and suppose that $F \in \mathbb{Z}[x_{1}, \ldots, x_{k}]$.
Can we always find a natural number $n(k)$ and $f_{1}, \ldots f_{n(k)} \in \mathbb{Z}[x]$ ...

**6**

votes

**3**answers

239 views

### Inverses in convolution algebras

Let $G$ be a locally compact totally disconnected group, and to make life easy let's suppose its Haar measure is bi-invariant. Let $C_c(G)$ be the space of locally constant complex functions on $G$ ...

**22**

votes

**6**answers

4k views

### Do convolution and multiplication satisfy any nontrivial algebraic identities?

For (suitable) real- or complex-valued functions f and g on a (suitable) abelian group G, we have two bilinear operations: multiplication -
(f.g)(x) = f(x)g(x),
and convolution -
(f*g)(x) = ...

**32**

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]."
...