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39
votes
5answers
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 ...
6
votes
2answers
527 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 ...
3
votes
0answers
181 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 ...
17
votes
1answer
3k 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 ...
12
votes
1answer
630 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 $m$ ...
2
votes
0answers
113 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): --- ...
32
votes
18answers
4k 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]." ...
14
votes
2answers
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 ...
17
votes
6answers
2k 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 ...
8
votes
1answer
334 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 ...
5
votes
2answers
1k 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.
3
votes
1answer
271 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, ...
3
votes
2answers
377 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
1answer
266 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 ...
1
vote
0answers
160 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 ...
3
votes
2answers
479 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 ...
2
votes
1answer
450 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 ...
1
vote
2answers
329 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 ...