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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Maps to quotient rings

If $R$ is a ring, $\mathfrak{a}$ is an ideal of $R$ and $S=A[x,y,z,\dots]$ where $A$ is a commutative ring, then is there a ring $S^\prime$ such that there is a bijection: ...
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The Binary Operation [closed]

Let $A$ is equal to $\{ 1,2,3 \}$. Binary Operator defined as $\star$ on $\mathcal P (A)$ by $X\star Y$ is equal to $(X-Y) \cup (Y-X)$ and also equal to $X\triangle Y$, where: $X,Y$ is a subset of ...
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Are there general conditions under which minimal generating sets can be expected to exist?

There exist algebraic structures $X$ with no minimal generating set. For example, $\mathbb{Q},$ viewed as an Abelian group. There also exist algebraic structures whose every generating set includes ...
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Number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$

For $p$ a prime, what is the number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$, $k \ge 1$? I can answer the $k=1$ case. For each element of $SL_6(\mathbb{Z}/p\mathbb{Z})$, there are ...
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a problem in homological algebra

For $C$ is abelian group satisfied $pC=0$ with p is a prime number and $G$ is abelian group. prove that $Ext_{Z}(C,G)\cong Hom(C,G/pG)$ I thought about this in 2 hours but couldn't prove it!
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Kaplansky characterization of principal Artin ring

I would like to learn the proof of this paper of Kaplansky where it is proven that for a commutative ring every module split as sum of cyclic module iff the ring is an Artin principal ideal ring (well ...
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Composition of module homomorphisms and their kernels

Let $R$ be a principal ideal ring and let $I$ and $J$ be two ideals of $R$. Suppose $\phi: R \times R \rightarrow R \times R$ and $\psi: R \times R \rightarrow R \times R$ are two $R$-module ...
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101 views

How can I solve $x^2+2=y^3$ in $\mathbb{Z}$?

Prove that $\left \{ (x,y)\in\mathbb{Z}^2:x^2+2=y^3 \right \}\subseteq \left \{ (-5,3),(5,3) \right \}$.
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Example of an nonidentity element in the kernel of the map.

This question is related to my previous question here. Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: ...
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Matrices as generators of free group.

In the introduction section of the paper Triples of $2\times 2$ matrices which generate free groups the authors mentioning some thing... In my words: The matrices $\begin{pmatrix}1 & 0 \\ 2 ...
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Inverse images of ideals

I was trying to solve the following exercise: Let $f\colon R\to S$ be a ring epimorphism, $I \subseteq S$ be an ideal, and $J = f^{-1}(I)$. Check that if $I$ is maximal (resp. prime) then $J$ is ...
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Equivalence classes of relation $\rho: (x,y)\in \rho \Leftrightarrow (\exists k \in \mathbb{Z})x-y=3k$

I don't understand how equivalence classes are $$C(1)=\{3k+1:k\in \mathbb{Z}\}$$ $$C(2)=\{3k+2:k\in \mathbb{Z}\}$$ $$C(3)=\{3k:k \in \mathbb{Z}\}$$ Could someone explain?
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Which of the algebra isomorphisms hold?

Fix $m, n \ge 1$. Which of the algebra isomorphisms below hold? $k\langle t_1, \dots, t_m\rangle \otimes_k k\langle s_1, \dots, s_n\rangle \cong k\langle t_1, \dots, t_m, s_1, \dots, s_n\rangle$ $k[ ...
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exercise Question-29 from contemporary abstract algebra [closed]

Consider the element A=(1101) in SL(2,R) what is the order of A? If we view A=(1101) as a member of SL (2,Zp), what is the order of A
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Is the fundamental weight basis (a.k.a Dynkin basis) an orthonormal basis?

The simple root $\alpha_i$ basis is not an orthonormal basis, as can be seen from the Cartan matrix, which encodes how much they aren't orthonormal. For simplicity, let's assume a simply-laced Lie ...
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For a prime integer $p \in \{2, 3, 5, \cdots\}$, is $pR$ a maximal ideal in $R$?

If $R$ is a commutative ring with unit and $p$ is a prime number ($2,3,5,\cdots$), then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$?
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66 views

How many ways can the group $\mathbb Z_5$ act on the set $\{1,2,…5\}$

How many ways can the group $\mathbb Z_5$ act on the set $\{1,2,...5\}$.What is the problem demanding ? How to approach it .Please give the way too approach it not the solution
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Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to ...
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Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
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25 views

Degrees of freedom of a complex vector space $V$ and its conjugate $\bar V$?

As an easy example consider the complex vector space $\Bbb C^2$. We can consider $\Bbb C^2$ as vector space over $\Bbb R$ and thus have the four basis vectors $$ \hat e =\{(1,0), (i,0), (0,1), ...
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Notation/definition problem for commutative binary operation

I'm trying to describe/define the commutative binary operation on a three-element set which when the operands are the same, gives the same element and when they are different gives the element which ...
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Problem book on linear algebra

Please refer a problem book on linear algebra containing the following topics: Vector spaces, linear dependence of vectors, basis, dimension, linear transformations, matrix representation with ...
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Automorphism of two members as Generator

Let $X=\langle a,b|a^{2^m}=b^{2^n}=1,[a,b]=a^{2^{m-1}}\rangle$, $m,n\ge 2$ If $\alpha \in Aut(X)$ (Automorphism Group of $X$) is defined as \begin{cases} \alpha(a)=a^{2^{m-1}+1}\\ ...
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Finiteness of subgroup $\rightarrow$ Finiteness of the group

Let $G$ be a group and $H$ be its abelian and normal subgroup. If $H$ is finite and maximal, prove that $G$ is finite. What I tried : Assume $H=\{e,h_2,\cdots,h_{n}\}$. As for each $j$, we ...
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48 views

Determine number of elements of order 12 of a group

Let's say we have a commutative group G that's specified by generators and relations. We find that the group G normal form is: $Z_2\times Z_6\times Z_{12}$ and that the elementary form is $Z_2\times ...
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Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. [closed]

I took this exercise for a long time but I can't prove it. Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. Anyone could help me?
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Can every ideal have a minimal generating set?

Let $I$ be an ideal of commutative ring $A$ with unity. Does $I$ have a minimal generating set? At times, I am able to compute what they are for specific example, but it seems like it is true in ...
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Why are algebras classified as being of a certain type?

In Grätzer's, Universal Algebra, page 33, Grätzer defines the concept of an algebra of type $\tau$, as follows: An algebra $\mathfrak{A} = \langle A ; F \rangle$ of type $\tau$ is a pair, where ...
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Nakayama automorphism $\sigma$ of Hecke Algebra $^0H^f_n$ is not inner for $n\geq 3$?

With $R=\mathbb{Z}[q_1,q_2]$, the Hecke algebra $H^f_n$ of $S_n$ is defined to be the $R$-algebra generated by $T_1,\dots,T_{n-1}$ satisfying $T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1}$, $T_iT_j=T_jT_i$ if ...
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Equivalence of two statements in a field.

I need to prove the following in order to prove something interesting about generalized quaternions: Let $K$ be a field and suppose $a \neq 0, b \neq 0$ are elements of $K$. Then the following are ...
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a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
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What is the smallest subfield of the complex numbers which has the property that every polynomial of odd degree has a root

It can be shown using the intermediate value theorem that every polynomial of odd degree with real coefficients must have at least one real root. I was just curious, are there any other smaller fields ...
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43 views

If $|xH|$ has order $n$, then there is an element $y$ with $|y|=n$ and $xH=yH$

Let $G$ be a group, and let $H$ be a normal subgroup with $|H|=m$. Suppose $n$ and $m$ are relatively prime. If $|xH|$ has order $n$, we wish to find an element $y$ with $|y|=n$ and $xH=yH$. It is ...
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Is there a method to calculate large number modulo?

Is there a (number theoretic or algebraic) trick to find a large nunber modulo some number? Say I have the number $123456789123$ and I want to find its value modulo some other number, say, ...
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Let $G$ be a compact group. If $\{a^n\}_{n \in \mathbb{Z}}$ is dense in $G$, then $G$ is abelian.

It was used in the middle of a theorem's proof and I am not sure how to prove this fact.
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54 views

Embedding tensor product of integral domains

Let $C$ be a subring of integral domains $A,B$ and let $C',A',B'$ denote their field of fractions respectively. Can we always embed $A\otimes_CB$ in $A'\otimes_{C'}B'$ by $a\otimes b\mapsto ...
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Minimal polynomial with repeated factors over an algebraically closed field.

Let $k$ be an algebraically closed field and let $V$ be a vector space over $k$ and let $T: V \to V$ be any linear transformation. I can't think of an example when the minimal polynomial of $T$ will ...
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70 views

Multiplication without exponents

Edited for clarity (The original statement of the problem I was explaining the problem as a redefinition of multiplication, as can see from the original comments; but I think the algorithmic ...
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Roots of unity over a general field $k$.

Let $k$ be an algebraically closed field of characteristic zero. Then $x^4-1$ factors linearly in $k$. Usually, I would consider the roots of $x^4-1=0$, the fourth roots of unity. Which in the case $k ...
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Why is there a $p\in \mathbb{N}$ such that $mr - p < \frac{1}{10}$?

I am reading the following part of the paper of Denef : Let $R$ be a commutative ring with unity and let $D(x_1,\dots , x_n)$ be a relation in $R$. We say that $D (x_1,\dots , x_n)$ is diophantine ...
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Show that the homomorph image of an abelian group is abelian

Since $G$ is abelian, we have that: $$ab = ba \implies \phi(ab) = \phi(ba) \implies \phi(a)\phi(b) = \phi(b)\phi(a)$$ Am I rigth?
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Can an element in a Noetherian ring have arbitrarily long factorizations?

Suppose $R$ is a Noetherian ring. Is it possible that an element $r\in R$ have arbitrarily long factorizations? That is, for all $n>0$, is there a factorization $r=a_{1n}a_{2n}\cdots a_{nn}$ such ...
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Roots of unity, intersection of fields [duplicate]

How to prove that intersection of $\Bbb{Q}(m)$ and $\Bbb{Q}(n)$ is equal to $\Bbb{Q}$, where $(m,n)=1$ and $\Bbb{Q}(n)$ is $n$-th cyclotomic field?
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Is there an isomorphism of additive groups between $\mathbb{Q/Z}$ and $\mathbb{Q}$?

I know that I have to study the order of every element in $\mathbb{Q/Z}$. But what do I do? I've been struggling of what to do for this question
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Definition of a peculiar quotient group of isometries

I just wrote a text file to sum up my ideas about the Riemann Hypothesis. This text is a draft, and I don't expect people here to say if this approach is interesting or not (but if by chance you think ...
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Primitive vectors in $A^n$

Let $A$ be a commutative ring with 1. Let n be a positive integer. I call a vector $(a_1,...,a_n) \in A^n$ primitive, if the ideal generated by $\{a_1,....,a_n\}$ is $A$. Question: Given a primitive ...
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Find the Kernel of the Homomorphic $f:\mathbb{Z}_8 \rightarrow \mathbb{Z}_4$

Previously I posted a question from "A Book of Abstract Algebra" to prove that the function, $f:\mathbb{Z}_8 \rightarrow \mathbb{Z}_4$ (shown below), is a homomorphism. $f = (0 \rightarrow 0, 1 ...
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If $\phi$ is an isomorphism, $\phi(g)^n = 1 \iff g^n = 1$. Doesn't this hold for homomorphisms too?

I need to prove that for an isomorphism $\phi$, the following is true: $$\phi(g)^n = 1 \iff g^n = 1.$$ We know that $$g^n = 1 \implies g\cdot g \cdots g = 1\implies \phi(g\cdot g \cdots g) = ...
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Proof that normalizer and center are subgroups

I've seen this proof for the center of a group $G$: $$C = \{x\in G:xg = gx \ \ \ \forall g \in G\}$$ So, the center is the set of all elements that commute with every $g$ of $G$. This subset of $G$ ...
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$G$ has order $p^a$, then the center of $G$ counts more than the identity

This question makes no sense to me. I don't know what it means by "counts more than the identity". Then, the exercise gives me a hint: "break G into equivalence classes of conjugacy elements and see ...