# Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify which topic of abstract algebra is most related to your question and also to help other users find similar questions through the search engine.

56,004 questions
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### Roots of a polynomial in an integral domain

Let $R$ be a ring and $f(X) \in R[x]$ be a non-constant polynomial. We know that the number of roots, of $f(X)$ in $R$ has no relation, to its degree if $R$ is not commutative, or commutative but not ...
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### Nil-Radical equals Jacobson Radical even though not every prime ideal is maximal?

Let's assume we have a commutative ring with identity. Can the Nil-Radical and the Jacobson Radical be equal in a non-trivial case (i.e. not every nonzero prime ideal in said ring is maximal)? Are ...
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### On the relationship between the commutators of a Lie group and its Lie algebra

I was trying to teach myself some basic Lie theory, and I came across this statement on Mathworld, relating the commutator of a group, $\alpha\beta\alpha^{-1}\beta^{-1}$, to the commutator of its Lie ...
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### Subgroups of finitely generated groups are not necessarily finitely generated

I was wondering this today, and my algebra professor didn't know the answer. Are subgroups of finitely generated groups finitely generated? I suppose it is necessarily true for finitely generated ...
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### The smallest normal set containing a subset $X$, what is a “normal set”?

I've been self studying abstract algebra from these notes. I've ran across the following Lemma: The confusing part is that $X$ is a "any subset" not any "subgroup", and that the term "normal set" ...
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### Examples of rings with idempotent elements

As a part of my studies in ring theory, I've encountered the concept of an idempotent element, i.e., an element $e$ such that $e^2=e$. Are there some interesting examples of rings with idempotent ...
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### Polynomial satisfying $p(x)=3^{x}$ for $x \in \mathbb{N}$

Let $p(x)$ be a polynomial with integer coefficients, which is not constant. Then is this condition possible: $$p(x)=3^{x}$$ whenever $x \in \mathbb{N}$. Motivation: https://mathoverflow.net/...
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### Sequence of Prime ideals in a Polynomial Ring

In the $d$ variable polynomial ring $R=k[x_{1},\cdots,x_{d}]$ show that $0, x_{1}R, (x_{1},x_{2})R, \cdots , (x_{1},x_{2},...,x_{d})R$ is a strictly increasing sequence of prime ideals and there is no ...
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### Applications of Logic and Algebra in Computer Science

I have used logic for specification of security policies in a security model. One of my reviewers asked a question that "why did you use logic and not algebra for this purpose, and what is your ...
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### Is there a name for this construction? (taking a pairing and obtaining a perfect one)

I believe that if we are given two modules $M,N$ over a ring $R$, and a pairing between them $M \otimes_R N \to R$, we can construct a perfect pairing $M'\otimes_R N' \to R$ by taking kernels. Is ...
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### What is the center of $V$, the Klein 4 group?

Please help - in my notes, it is the group $V$ itself. I just want to confirm this. Can you also explain and give an example if that is possible?
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### Showing $G$ is the product of groups of prime order

Let $G$ be a (not necessarily finite) group with the property that for each subgroup $H$ of $G$, there exists a `retraction' of $G$ to $H$ (that is, a group homomorphism from $G$ to $H$ which is ...
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### Coloring the faces of a hypercube

I will restate the 3-D version of the problem. In how many ways can you color a regular cube with 2 colors up to a rotational isometry. The answer is of course a special case of Burnsides Lemma which ...
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### Number of Elements of order $p$ in $S_{p}$

An exercise from Herstein asks to prove that the number of elements of order $p$, $p$ a prime in $S_{p}$ is $(p-1)!+1$. I would like somebody to help me out on this and also i would like to know ...
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### Quantum Jordan Algebras(Still wondering!)

I have attempted to google for this, but the searching is marred by the close relationship of Jordan Algebras and Quantum Mechanics. I have been passively thinking of this question for ages, tonight I ...
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### Why is $\mathbb{Q}(t,\sqrt{t^3-t})$ not a purely transcendental extension of $\mathbb{Q}$?

This question is taken from Dummit and Foote (14.9 #6). Any help will be appreciated: Show that if $t$ is transcendental over $\mathbb{Q}$, then $\mathbb{Q}(t,\sqrt{t^3-t})$ is not a purely ...
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### What are the most special spaces between which rigid transformations preserve the structures of the spaces

An affine transformation is a linear transformation followed by a translation. They are morphism between affine spaces. A rigid transformation consists of a rotation and a translation. I was ...
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### Relationships between Symmetric Groups

Here is a question that I can't find the answer to in my notes or my textbook. Is there any relation between $\text{Sym}_n$ and $\text{Sym}_k$ where $k < n$? Is $\text{Sym}_k \subset \text{Sym}_n$?...
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### Is there a particular characterization of $Aut(\mathbb{Q}^3)$?

I was considering the group of automorphisms on a vector space $\mathbb{Q}^3$ with the binary operation of addition. Is there a general description of the elements of this group? At first, it seemed ...
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### Quotienting a Set by an Equivalence Relation such that the Natural Projection is a Homomorphism

When studying group theory, I was interested to learn that a the quotient set $G/N$ of a group $G$ is again a group with $\pi\colon G\to G/N$ a homomorphism when $N$ is a normal subgroup of $G$. Is ...
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### Reducing Relations to Commutator Relations

I think this is obvious and I am just missing something but: Given some finitely presentable algebra, can we always reduce the relations to commutator relations? And moreover, if yes, then is ...
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### Can two structures be embeddable in each other, but not isomorphic?

I was reading about isomorphisms and homomorphisms on general structures, and first came across the definition of an injective homomorphism, or an embedding. This made me curious, is it possible for ...
We know this is true for commutative ring, but if $S\subset R$ is a left and right Ore set, and $S^{-1}R$ its localization by this Ore set, is this always a flat $R$-module?