Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Atiyah-Macdonald Exercise 2.15

I have worked out a solution to exercise 2.15 of Atiyah-Macdonald, which is needed in the solution of 2.3 (see Atiyah-Macdonald 2.3). However, the solution seems overly complicated, and I am not ...
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When is a power of $m$ cycle is also an $m$ cycle?

I have a question taken from Abstract Algebra by Dummit and Foote ($pg.33$ $q.11$): Let $\sigma\in S_{n}$ be an $m$ cycle. Show that $\sigma^{k}$ is also an $m$ cycle iff $(k,m)=1$ My efforts: By ...
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Automorphisms of $\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p$

Consider the abelian group $$G = \underbrace{\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p}_{n},$$ where $p$ is prime and $1\le n \le p$. I want to show that $G$ has no automorphism of order $p^2$. I ...
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product of comprime numbers and UFD

It is well-known that if a product of coprime numbers is a perfect square, so are the numbers. The proof depends on fundamental theorem of arithmetic, and this implies that in a UFD, if ab is a ...
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1answer
39 views

Noetherian ring with finite dimensional vector space structure

Let $K$ be a field and $R$ a ring with finite dimensional vector space structure over $K$. Is $R$ necessarily a Noetherian ring? If $K \subset R$, then any ideal in $R$ is also a subspace and, since ...
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49 views

Normal subgroup created by a bunch of elements

if I have a finite group $G$ and a bunch of elements that are the elements of a set $A$. How can I systematically calculate the smallest normal subgroup of $G$ that contains $A$? I am rather ...
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33 views

Little help in this theorem from Fulton

I'm studying the Fulton's algebraic curves book and I have the following doubts in the end of the page 9: I didn't understand why the following equations hold: $$I\left(\bigcup_i ...
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61 views

Must the centralizer of a non-identity element of a group be abelian?

Q1: Must the centralizer of a non-identity element of a group be abelian? I challenge everyone by asking further about this question one step further. The definition of centralizer is: Let a be ...
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1answer
27 views

A two sided ideal of a Noetherian ring

$R$ is a left Noetherian ring with a minimal left ideal. Consider the set of minimal left ideals of $R$ ordered by inclusion. Then there is a maximal element $\mathfrak b= \bigoplus_{i\in I} \mathfrak ...
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565 views

What's the name of this algebraic property?

I'm looking for a name of a property of which I have a few examples: $(1) \quad\color{green}{\text{even number}}+\color{red}{\text{odd number}}=\color{red}{\text{odd number}}$ $(2) \quad ...
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2answers
102 views

A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
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Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
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Atiyah-Macdonald 2.3

In solving question 2.3 from Atiyah & Macdonald's commutative algebra textbook, I run into the following difficulty: Let $A$ be a local ring with $k:= A/mA$ its residue field and let $M$ and $N$ ...
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1answer
38 views

Show that $D \cong {\rm End}_A(D^n)$ where $D$ is a division algebra and $A\cong M_n(D)$

Define $A$ to be a finite dimensional simple algebra over a field $k$. $D$ is a $k$-division algebra (not necessarily commutative) such that $A\cong M_n(D)$ for some integer $n$. Let $L$ be a minimal ...
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42 views

Is this sequence of presheaves exact?

On p.298 of his Homological Algebra text, Rotman considers the sequence of presheaves on $X=\mathbb{C}-\{0\}$ : $0 \to \mathbb{Z}\to \mathcal{O}\to \mathcal{O}^\times \to 0$ where $\mathbb{Z}$ is the ...
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18 views

quasiprimitive non-solvable groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
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1answer
42 views

Can every group be faithfully represented as a group of permutations?

Definition (Group action) An action of a group $G$ on a mathematical object $X$ is a group homomorphism $G \rightarrow \mathrm{Sym}(X)$. i.e. Given an action $f$ of a group $G$ on a mathematical ...
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31 views

My question is about the definition of a map called the “reduction map”.

Let $G$ be a group and $N$ normal in $G$. I have read about a map $\alpha : G\rightarrow \frac{G}{N}$ called the reduction map mod $N$. I would love if someone could please explain this to me. Is it ...
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1answer
28 views

Lie ideals of $gl_n(K)$

I am looking for some reference where I can find a detailed study of the Lie ideals of the general linear Lie algebra $gl_n(K)$ with the bracket $[A,B]=AB-BA$, where $K$ is a field (if there are ...
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1answer
45 views

A not free $\mathbb{Z}$-module .

From this post If $\{M_i\}_{i \in I}$ is a family of $R$-modules free, then the product $\prod_{i \in I}M_i$ is free? I see that $\mathbb{Z}^{\mathbb{N}}$ is not free, someone can explain me why? ...
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1answer
34 views

Does this property of normal subgroups have a name?

I've recently started studying abstract algebra and I noticed that a subgroup N is normal iff $a \equiv x$ and $b \equiv y$ implies $ab \equiv xy$ mod N for any a,b,x,y. Is this an important ...
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2answers
35 views

Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
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87 views

Introduction and Prerequisites to Abstract Algebra

So I've seen similar questions asked, but none that really helped me out. I'm going to be a freshman in college next year, having already taken Multivariate Calculus and Elementary Linear Algebra. Of ...
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88 views

Group of order 1183 is abelian if and only if contains an element of order 91

Let $G$ be a group such that $|G|=1183=7\cdot 13^2$. Show that $G$ is abelian if and only if $G$ has an element of order $91=7\cdot 13$. What i did: $7||G|\Rightarrow \exists x\in G : |x|=7$ and ...
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Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
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39 views

Find a multiplicative inverse of an element in a field

Suppose we have an element $\sigma=p+qa\rho+rd\rho^{-1}\in K$ where $K=\mathbb{Q}(\rho)$ where $[K:\mathbb{Q}]=3$ I want to find a multiplicative inverse of $\sigma$ i .e ...
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119 views

If for any two principal ideals one contains another, then for any two ideals one ideal contains another

Let $R$ be a commutative ring with identity. Assume that for any two principal ideals $Ra$ and $Rb$ we have either $Ra\subseteq Rb$ or $Rb\subseteq Ra$. Show that for any two ideals $I$ and $J$ in ...
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1answer
44 views

A Direct Sum of Members of a Certain Class of Modules

Let $S$ be a class of $R$-modules and let an $R$-module $M$ be countably generated. Suppose that, for every direct summand $K$ of $M$, each element of $K$ belongs to a direct summand of $K$ that is ...
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For extension fields, does $[F(a,b):F(a)]=[F(b):F]$?

sorry if this question seems obvious, For a field F and $a,b\notin F$, does $[F(a,b):F(a)]=[F(b):F]$? If so, how do you prove it or is there a counter example?
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Show that the image of a linear transformation is equal to the kernel

Let $\phi$ be a linear transformation such that $\phi: V\to V$ We are given the following facts: $\dim(V) = 8$ $\dim(\mathrm{Im}(\phi)) = 4$ $\phi\circ\phi=0$ Show that $\mathrm{Im}(\phi) = \ker ...
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48 views

A Question about the Proof of Eisenstein's Irreducibity Criterion

Statement: Let $f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_0 \in \mathbb Z[x]$. If there is a prime $p$ such that $p \nmid a_n, p \mid a_{n-1}, \dots,p \mid a_0$ and $p^2 \nmid a_0 $, then $f(x)$ ...
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primitivity of a polynomial over a field

Suppose that we have a polynomial $f(x)=ax^3-bx^2+cx-d\in\mathbb{Z}[x]$ then $f(x)$ will be called primitive if $(a,b,c,d)=1$ I have been told that over a field $F$ there is no notion of ...
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What is an advatage of defining $\mathbb{C}$ as a set containing $\mathbb{R}$?

It is a theorem that every field with least upper bound property and Archimedean property is isomorphic to each other. So it seems not necessary to define $\mathbb{R}$ exactly and we simply denote ...
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A condition of equivalence of flatness and projectiveness

This is a problem in "Foundations of Module and Ring Theory" of Wisbauer: " Let $R$ be a subring of the ring $S$ containing the unit of $S$. Show that a flat $R$-module $N$ is projective if and only ...
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Question on the definition of sheaves.

When defining a sheaf of $O_X $-modules, or sheaves in general, I have nearly always seen it given as a functor from the category of all the open sets to another category with the usual properties. ...
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When does $n$-dimensional algebra have $m$-dimensional faithful representation?

Suppose we have an $n$-dimensional associative unital algebra $A$ over a field $k$ (assume $\operatorname{char}(k)=0$ and maybe even $k$ is closed). I would like to know what is the minimal ...
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1answer
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Prove $Q_8$, the group generated by two complex matrices $A$ & $B$ (see below) is a nonabelian group of order 8.

Problem: Let $Q_8$ be the group (under ordinary matrix multiplication) generated by the complex matrices $A=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $B=\begin{pmatrix} 0 & i \\ i ...
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1answer
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Please check my proof on: $\sim$ is an equivalence relation $\Leftrightarrow S<G$

Problem: Let $\emptyset\ne S\subset G$, where $G$ is a group, and define a relation on $G$ by $a\sim b\Leftrightarrow ab^{-1}\in S$. Show that $\sim$ is an equivalence relation if and only if $S$ is a ...
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Defining an inner abstract vector space

Since an inner product space is an abstract vector space with an additional structure called an inner product, and this additional structure is a component wise operation that associates each pair of ...
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19 views

Embedding the base ring in the augmentation ideal of a group algebra

Let $G$ be a finite group. Then the group algebra $\mathbb{Q}G$ trivially contains $\mathbb{Q}$. But when (i.e. for which $G$) does the augmentation ideal $I_G=\{\sum_{g\in G} r_g\,g \mid \sum_{g\in ...
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Prove $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are finite.

Prove that $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are finite. What I've done Use the definition of subgroup: $G$ is a group and $H \subseteq G$. $H \le G \iff HH=H $ and ...
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1answer
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Question concerning how a map extends to a homomorphism.

Let $G$ be a finite group with finite set of generators $g_1,g_2,...$ that is not known "explicitly" as a group, but rather indirectly as a faithful representation as a subgroup of $S_n$. If I define ...
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Quadratic form using definiteness of a matrix. [on hold]

Show that the set {x: x'Ax<1} is bounded if and only if A is positive definite.
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Determining a spanning set for $X/\bigcap_{i=1}^N \ker{\lambda_i}$, where each $\lambda_i$ is a linear functional on $X$

Let $X$ be a vector space over a field $K$. Suppose that $\{\lambda_i\}_{i=1}^N$ is a collection of linear functionals $\lambda_i : X \to K$. Let $W$ be the subspace $\{ x \in X \mid \lambda_i(x) = ...
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2answers
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Identifying the algebra

In order to solve an obscure (physics) problem I have been considering whose details are not important, I am looking for elements (I am thinking in terms of matrices and their products but this may ...
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Prove that an expression is zero for all sets of distinct $a_1, \dotsc, a_n\in\mathbb{C}$

A while ago one of my professors gave the class a problem "to think about when lying on the beach." Well, I've been on the beach several times since then to no avail and my curiosity has finally ...
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If $H$ and $K$ are subgroups of G then $H \times K$ is a subgroup of $G \times G$

I know that if $H$ and $K$ are subgroups of $G$ then $HK= \{ hk \mid h \in H , k \in K\}$ is not necessarily a subgroup of $G$, this requires that $HK = KH$. But it follows that if $H$ and $K$ are ...
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A city wants to encourage downtown

could you please help me with this ( part d ) A city wants to encourage downtown employees to use public transportation. Below is the time in minutes to get to work on one morning according to ...
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Prove $\alpha: \mathbb{Z_n} \rightarrow \mathbb{Z_n}$ defined by $\alpha(s)=rs\mod{n}$ is an automorphism.

I'm working on proving the following claim: "Let $r \in U(n)$ and $\forall s \in \mathbb{Z_n}$, $\alpha: \mathbb{Z_n} \rightarrow \mathbb{Z_n}$ defined by $\alpha(s)=rs\mod{n}$ is an automorphism." ...
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A problem on matrices over a commutative ring

Let $M_{m,n}(R)$ denote an $m\times n$ matrix with each entry over a commutative ring $R$, $m\leq 2\leq n$, and there is a matrix $\mathbf{B} = M_{m,n}(R)$. $\mathbf{B}\mathbf{s} = \mathbf{a}$, where ...