Tagged Questions

Do NOT use this tag. Instead you could consider gr.group-theory, ac.commutative-algebra, ra.rings-and-algebras or various more specific tags.

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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 ...
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$A$ 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 $A \not\cong A \oplus \mathbb{Z}$?
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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 ...
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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 ...
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$R$ 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 ...
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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]." ...
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Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
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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 ...
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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) = &...
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Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$ such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$....
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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.
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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 ...
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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 $\mathfrak{S}_n$ ...
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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 ...
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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 ...
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Formally undecidable problems on finitely presented quandles

In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite ...
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A weird question about two weird decompositions of $\mathbb{R}$ as a $\mathbb{Q}$-vector space

While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers: Question: Do there exist a non-trivial ...
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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 ...
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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 $m$ ...
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Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...
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$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group. Denote the ...
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Why are ring actions much harder to find than group actions?

I admit freely that the following question is a bit of a fishing expedition inspired by this lovely "definition" of a module as found on Wikipedia: A module is a ring action on an abelian group. ...
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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 ...
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Linearly independent subsets of a free module

Is it true that the cardinality of every maximal linearly independent subset of a finitely generated free module $A^{n}$ is equal to $n$ (not just at most $n$, but in fact $n$)? Here $A$ is a nonzero ...
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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 ...
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For the past one week, I have been trying to learn more about Automorphism groups of different groups. Very recently one of my friend asked this question to me: What is the Autmorphism group of $(\... 4answers 383 views Real and Quaternionic Representations according to Weights According to this question, it is easy to know whether a representation is self dual or not: just check if the weight distribution in space is symmetric about the origin. Now, for self dual ... 1answer 636 views Elementary proof for Hilbert's irreducibility theorem I have tried to find a complete proof for Hilbert's irreducibility theorem, but everything I found was way beyond my level of understanding. I am only interested in the simple case where the ... 5answers 6k views Does category theory help understanding abstract algebra? I'm studying category theory now as a "scientific initiation" (a program in Brazil where you study some subjects not commonly seen by a undergrad), but as I've never studied abstract algebra before, ... 2answers 822 views Free division rings? Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators? 1answer 363 views Given a rational number a/b does there exist a finite group G and an automorphism f s.t. f maps exactly a/b elements of G to their own inverses? I was helping a friend prepare for his intro abstract final and he mentioned the professor had once asked the question name a group and an automorphism that takes 3/4 of the elements of the group to ... 1answer 578 views A group whose automorphism group is cyclic Is there an Abelian group$A$which is not locally cyclic whose automorphism group is cyclic ? This question was first posted here. 1answer 575 views Is$\mathbb{R}$a$\mathbb{C}$-module without AC? Assuming ZFC. We can make$(\mathbb{R},+)$into a nontrivial(scaler multiplication is not identicaly zero)$\mathbb{C}$-module. Now my questions are? 0.Is it consistent with$ZF$that$\mathbb{R}$is ... 2answers 1k views Galois group of a product of polynomials How can I compute the Galois group of the polynomial$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. 10answers 8k views Learning Algebra & Group Theory on my own [closed] I'm learning Algebra & Group Theory, casually, on my own. Professionally, I'm a computer consultant, with a growing interest in the mathematical and theoretical aspects. I've been amazed with ... 1answer 426 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 ... 0answers 160 views $A$-module is free if and only if equation involving Hilbert-Poincaré series holds,$M$infinitely generated case See my question here. Let$A = \oplus_{i \ge 0} A_i$be a nonnegatively graded commutative algebra and$M$a nonnegatively graded$A$-module. Assume in addition that$A_0 = k$and all vector ... 4answers 648 views Groups and rings which are not sets An algebraic structure such as a group, ring, field, etc. is usually defined to be a set with some operations satisfying certain properties. I am curious what, if anything, goes wrong when the ... 1answer 263 views $k[x_1, \dots, x_n]$is free iff$\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$isomorphism? For any subgroup$G \subset \text{GL}_n(\mathbb{R})$the set$\mathbb{C}[x_1, \dots, x_n]^G$, of$G$-invariant polynomials, is a graded subalgebra of$\mathbb{C}[x_1, \dots, x_n]$, resp. the set$\...
I am trying to move in categorical algebra beyond the basics. A Lawvere theory L is a small category with finite products. (I know that there also is a functor $(skeleton(FinSet))^{op}\to L$, which ...