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

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Prove a bijection of ideals exists.

A problem from Intro to Abstract Algebra by Hungerford If $f:R\rightarrow S$ is a surjective homomorphism of rings with kernel $K$, prove that there is a bjective function from the set of all ideals ...
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Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups?

Well, this is my question. Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups (maybe one being trivial)? Thanks!
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isomorphic quotient rings?

I have trouble in determining, whether two rings are isomorphic: Let's have $R = GF(3)$ and rings $R[x]/(x^2+x+2)$ and $R[x]/(x^2+2x+2)$. How can one determine whether these two rings are ...
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Sites or youtube videos to learn algebraic geometry

Is there any sites or free lecture videos to learn algebraic geometry? or should I call abstract algebra? I want to understand about rings, ideals, and real spectrum of rings but my understanding on ...
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Real Spectrum of a Commutative Ring

Could anyone explain briefly what is a real spectrum of a commutative ring? If it possible please give me some easy example. Many thanks in advance. $\text{Spec}_{r}(A)$
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Find all elements of quotient ring

I am studying the definitions of rings, ideals, and quotient ring, but I have a bit problem to apply the theory into the practice. I would like to find all elements of quotient ring $\mathbb{Z}[i]/I ...
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Can we find some constraint about order of $xy$ in a group $G$?

Can we determine order of $xy$ in $G$ if we know order of $x$ and $y$ ? I know that answer is yes for abelian groups and I guess the answer is no for nonabelian case. That is why I am looking for ...
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Normal Subgroups with factor groups

In part ii) I understand $(Nx)^2=Nx^2=N $ $(Nx \text{ order }2)$ but I do not understand why this implies $x^2\in N$
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How do i prove that $\gcd(a_1,\ldots,a_n)\operatorname{lcm}(a_1,\ldots,a_n)=a_1\cdots a_n$?

Let $a_1,\ldots,a_n$ be nonzero integers. Define $$G=\{A\in\mathbb{Z}:A \text{ is a linear combination of } a_1,\ldots a_n\}$$ My definition for $\gcd(a_1,\ldots,a_n)$ is the principal ideal of $G$. ...
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Examples of Cohen-Macaulay rings.

I've just started to learn about Cohen-Macaulay rings. I want to show that the following rings are Cohen-Macaulay: $k[X,Y,Z]/(XY-Z)$ and $k[X,Y,Z,W]/(XY-ZW)$. Also I am looking for a ring which is ...
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Necessary and sufficient for $\operatorname{orb}(x)=\operatorname{orb}(y) \iff \operatorname{Stab}(x)=g\operatorname{Stab}(y)g^{-1}$

Are orbits equal if and only if stabilizers are conjugate? You may get some insights from the link above. My Question: What is the necessary and sufficient condition for the above statement to be ...
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Meaning behind the conjugacy class in describing geometry of solids

When we consider the group of rotataional symmetry , say of a cube or a dodecahedron, it is not difficult to see the symmetry group is isomorphic to a $S_4$ , $A_5$ respectively. Mooreover, when you ...
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The group of rigid motions of the cube is isomorphic to $S_4$.

I want to solve the following exercise from Dummit & Foote. My attempt is down below. Is it correct? Thanks! Show that the group of rigid motions of a cube is isomorphic to $S_4$. My ...
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95 views

Every radical is prime?

$a$ is an ideal of $A$. $$f:A\to A/a,\ \ x∈r(a)$$ r(a) is a prime ideal? proof 1: $x^n\in a$ for some $n \Rightarrow (x+a)^n\in a$ for some $n \Rightarrow f(r(a))=\text{nil-radical}$ in $f(a) ...
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Question about Principal Ideals

I'm just learning basic ring theory and had a question about the definition of a principal ideal. For a commutative ring $R$ with unity, Fraleigh defines the principal ideal generated by $a\in R$ as ...
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~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
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Can we find an element of infinite order in a symmetric group of infinite order?

In particular, I'm thinking of a simple example: the group $S_\Omega$ given $\Omega = \{1, 2, 3, ...\}$. I've been thinking of elements of $S_\Omega$ in terms of their cycle decomposition, which may ...
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Jucys-Murphy elements confusion

I am taking a class called "Harmonic Analysis on Finite Groups" and am studying for an exam. We have recently been talking about the representation theory of the symmetric group (over $\mathbb C$). ...
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(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
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2answers
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(Counting problem) very interesting Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ (1) \quad x_1 y_2 \equiv x_2 y_1 \pmod{N}\\ (2) \quad x_1 y_3 ...
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Dummit and Foote page 512 claim

Dummit and Foote Abstract Algebra page 512 Given any field F and any polynomial $p(x)\in F[x]$ one can ask a similar question: does there exist an extension K of F containing a solution of the ...
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Localization of a module as direct limit

Let $A$ be a commutative ring, $S \subset A $ a multiplicatively closed set and $M$ an $A$-module. For every $s \in S$ we denote by $M_{s}$ the localization of $M$ with respect to $\{ 1, s, s^2, ...
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Prove this morphism is a derivation

Let $A$ a ring, $B$ a $A$-algebra, $I=\ker (B\otimes_A B \to B)$ given by multiplication, i wanna see why is $$d:B \to I/I^2$$ $$b \mapsto b\otimes 1 -1 \otimes b$$ well defined (in fact, why mod ...
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Is this homomorphism surjective?

For part b) I understand how to show there is a homomorphism from D2p to H via the method described, but why can we say this homomorphism is surjective?
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Can someone explain to me this answer about subrings?

so I know how to prove that $\mathbb{Z}\left[\sqrt{2}\right]=\{a+b\sqrt{2}:a,b\in\mathbb{Z}\}$ and $\mathbb{Z}\left[\sqrt{3}\right]=\{a+b\sqrt{3}:a,b\in\mathbb{Z}\}$ are subrings of $\mathbb{R}$. ...
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Is $i \in \mathbb{Q}[\sqrt[4]{-2}]$?

I have a homework question from Artin's Algebra that asks Is $i \in \mathbb{Q}[\sqrt[4]{-2}]$? I suspect that this is not true because $i \sqrt{2} \in \mathbb{Q}[\sqrt[4]{-2}]$ and $\sqrt{2}$ is ...
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Surjective Homomorphism

For $2$ i get that $C_2 \times C_2$ is not cyclic and I understand that if the homomorphism is surjective it must cover the entirety of $C_2 \times C_2$, but i don't follow why the image must be ...
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50 views

Isomorphism preserves exactness

Let $R$ be a commutative ring with unity. Let $A_i$ be an R-module for every $i$. Consider a sequence of modules $$\xrightarrow{\delta_{i-1}}A_{i-1}\xrightarrow{\delta_{i}} ...
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About a class of commutative rings that they have maximal ideals for any element non-inversible in $ZF\neg AC $: resolution

Let $\mathcal{N}{oetherian}\mathcal{C}\mathcal{R}{ng} \overset{def}{=} {\left\lbrace{ R \in \mathcal{C}\mathcal{R}{ng} \wedge R \,\text{is}\, \mathcal{N}{oetherian} }\right\rbrace}$. I define the ...
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Betti numbers and Fundamental theorem of finitely generated abelian groups

My textbook (author : fraleigh) says that Fundamental theorem of finitely generated abelian groups Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of ...
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multiplicative inverse in factor ring

If I need to find the multiplicative inverse of an element in some $T[x]/(m)$ factor ring, do I need to solve a diophantine equation to get the solution? Let the element be $f$. Then $fu \equiv 1$ ...
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Why is the group G a normal subgroup of itself?

Apparently $g^{-1}Gg=G$ for all $g$ in $G$. I understand that by closure if you multiply on the right by $g$ and the left by $g^{-1}$ you will get an element of the group $G$. However, how do i know ...
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Is there a simple proof for Fundamental theorem of finitely generated abelian group?

I'm studying two abstract algebra texts simultaneously now, namely, 'Dummit&Foote' and 'Fraleigh'. Both of these texts introduce 'Fundamental Theorem of finitely generated abelian group' without ...
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Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
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differentials on formal schemes

Let $A$ be a topological ring, we say that it is pseudo-compact if : there is family of ideals $\Lambda_A$ which gives a basis of neighborhoods of $0$ and such that $A/\mathfrak{a}$ is artinian for ...
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121 views

$k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$

This is part of an exercise from Eisenbud: $k$ is a field, describe as explicitly as possible a) $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$ b) $k[x] \otimes_{k} k[y]$ Any hint ?
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Do normal group endomorphisms form a normal submonoid?

What it says on the tin. A group endomorphism $v\colon G\to G$ is called normal if $v(aba^{-1})=av(b)a^{-1}$ for all $a,b\in G$. Equivalently, the map $g\mapsto v(g^{-1})g$ is a group homomorphism. ...
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51 views

Topologically dense subgroup

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
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Question on HSP and SHPS inquality.

In the screenshots attached above George Bergman outlines his way of proving $HSP \ne SHPS$ I understand the first definition as the group of affine transformations and each element of the group ...
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One to one mapping from $A(S_1)$ into $A(S_2)$

Let $S_1$ and $S_2$ be two sets. Suppose there exists a one to one mapping $\phi$ of $S_1$ into $S_2$. Show that there exists an one to one mapping from $A(S_1)$ into $A(S_2)$, where $A(S)$ means ...
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Degree of splitting field of $X^n-1$ over $F_p$

Suppose that $n$ is a natural number and p is a prime that does not divide n. Let $L$ be the splitting field of the polynomial $X^n-1$ over $\mathbb{F}_p$. Show that $[L:\mathbb{F}_p]$ is the smallest ...
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Degree of closure of $\mathbb{Q}_p$

In order to prove that algebraic closure of $\mathbb{Q}_p$ is infinite, I took the polynomial $x^n-p$ with $n>1$ over $\mathbb{Q}_p$ to show that this eqaution has no solution for infinite cases to ...
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An Algebraic Version of vector spaces

Consider the following set of real numbers $\mathcal{X}=\{1,2,3,\sqrt{2}+1,\pi+\sqrt{2}\}$. Lets consider the set of all linear combinations with integer coefficients of these numbers which I will ...
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how every simple and noabelian group will be not nilpotent?

this question arise when I was searching about tarski monsters http://en.wikipedia.org/wiki/Tarski_monster_group.this group is nilpotent because it is simple and noabelian,now I don't know how it is ...
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Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, ...
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Semidirect products of cyclic groups

Consider $A=\langle a\rangle$, cyclic group of order $9$ and $B=\langle b\rangle$, cyclic group of order $3$. Consider now the following action of $B$ on $A$ via automorphism: ...
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Working towards Abel's proof of unsolvability of quintics

I am currently doing a course in Abstract Algebra. I have been told that while some of the basic theory is laid down, we will not get as far as actually proving the unsolvability of quintics. ...
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2D Rubik's cube?

There is a $3\times3$ matrix filled by numbers 1~9 that might look like this $$\begin{bmatrix}3 & 8 & 2 \\ 4 & 1 & 6 \\ 7 & 5 & 9\end{bmatrix}$$ All its rows and columns can ...
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if $G' <H < G$ then $H$ is normal in $G$.

if $G' <H < G$ then $H$ is normal in $G$. ($G'$ is the commutator subgroup of $G$.) This is what I do: because $G' < H$ we have $\frac{H}{G'} \triangleleft \frac{G}{G'}$. because ...
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Group under composition $\circ$?

Define the set of all affine real-valued functions $G:=\{f_{a,b} \mid a,b \in \mathbb{R},a \neq 0\}$ where $f_{a,b} : \mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_{ab}: x \mapsto ax+b$. Is ...