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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Logic problem: Atiyah-Macdonald 1.11

Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following: "Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ ...
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0answers
15 views

Internal Direct Sum [closed]

Let F be a field and V a vector space over F. Let $W_1$, $W_2$, U be proper subspaces of V. Assume that each $W_i$ is a complement of U, that is $W_i \oplus U=V$ is an internal direct sum. Do not ...
3
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1answer
25 views

Permutations Isomorphic to $S_4$

Prove that the group generated by permutations $(0 2 6 4)(1 3 7 5)$ and $(4 2 1)(6 3 5)$ are isomorphic to the symmetric group $S_4$. I approached this problem by labeling the vertices of a cube. ...
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1answer
16 views

Equivalence for binary quadratic forms with positive square discriminant

I recently encountered an interesting proposition without proof: If $f(x,y)$ is a quadratic form whose discriminant is a non-zero perfect square, then $f(x,y)$ is equivalent to a form $a*x^{2} + ...
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3answers
34 views

Find a non-trivial semidirect decomposition of the following groups

Find a non-trivial semidirect decomposition of the groups $S_n$, $n \geq 3$, $D_{2n}$, $n \geq 3$ and $A_4$. Prove that $A_n$, $n \geq 5$ and $Q_8$ have no non-trivial semidirect decompositions. How ...
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votes
1answer
11 views

torsion of a module and exactness

Given a PID $A$ and $A$-modules $M$, $M'$, and $M''$. Assume that $$0\rightarrow M'\xrightarrow{f} M\xrightarrow{g} M''$$ is an exact sequence then prove that $$0\rightarrow ...
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1answer
21 views

$A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$

I've a question from gallian which states: Show that $A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$.Generalize.What happens if $3x$ is replaced by $4x$... ...
2
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4answers
62 views

$\mathbb R[x]/\langle x^2+1\rangle$ is a field

How to show that $\mathbb R[x]/\langle x^2+1\rangle$ is a field. I wrote the representation of $\mathbb R[x]/\langle x^2+1\rangle$ =$\{a+bx+\langle x^2+1\rangle\big|a,b\in \mathbb R\}$. Now how ...
0
votes
2answers
31 views

Units and nilpotents in quotient ring. [closed]

$A$ is a commutative ring and $N(A)$ is the nilradical of $A$. If $A/N(A)$ is a field, show that every $a \in A$ is invertible or nilpotent.
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1answer
31 views

Krull dimension of a finitely generated integral domain over $k$ is equal to the transcendence degree.

This theorem is from Matsumura (p.34) Let $k$ be a field and $A$ an integral domain which is finitely generated over $k$. Then $\dim A = \operatorname{trdeg}_k A$ (where $\operatorname{trdeg}_k ...
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1answer
20 views

Homomorphism between 2 abelian groups sending one given element to another given element

Let $G$ and $G'$ be arbitrary abelian groups. Fix a $g \in G$ and $h \in G'$. Then does there exist a homomorphism $\phi$ such that $\phi(g) = h$?
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1answer
34 views

Doubt regarding zero elements in factor ring :$\mathbb Z[i]/\langle3-i\rangle$

I have the factor ring $\mathbb Z[i]/\langle3-i\rangle$ and am asked to find elements zero in this ,they are $0,3-i,i(3-i),(3-i)+i(3-i)$. But I can't understand how do we guarantee these are the only ...
2
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3answers
61 views

Proving ring $R$ with unity is commutative if $(xy)^2 = x^2y^2$

I tried several methods to solve this but couldn't get through. Now the solution in almost all the textbooks goes like this. First take $x$ and $y+1$ so that $ (x(y+1))^2 = x^2(y+1)^2 => xyx = ...
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1answer
53 views

What is meant by a kernel and homomorphism in algebraic structures?

I have just started with discrete maths. I was doing some group theory and I stumbled upon kernel and homomorphism. I didn't understand what was written in the book. I googled it up and also looked ...
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0answers
35 views

Tensor product of R-algebras

I'm dealing for the first time with tensor product of $R$-algebras $S,T$ (where $R$ is a commutative ring). $S\otimes_{R}T$ Can someone explain me what is the advantage given by the fact that $S$ ...
0
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1answer
21 views

Isomorphisms imply other isomorphisms

Let A, B, and C be vector spaces. If A is isomorphic to C, and B is isomorphic to C, do these isomorphisms imply that A is isomorphic to B? Does it suffice to say that the result follows from ...
2
votes
1answer
42 views

Problem regarding a homomorphic mapping

I can't understand this: suppose we have a homomorphism $\phi :G \to G'$ such that it induces these two maps : $$\phi_*:\tau\to \tau'$$ ,a map from subgroups of G to subgroups of G' s.t ...
3
votes
1answer
45 views
+50

How to define a taxonomy of non associative operations?

Let $A$ be a set, and let $a,b,c\in A$. Let also $\circ: A\times A\rightarrow A$ be a binary operation on $A$. We agree as usual to write $a\circ b$ to mean $\circ(a,b)$. We say that $\circ$ is ...
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1answer
22 views

An Ideal which is Maximal additive subgroup is a Maximal Ideal

How should I prove this: Any Ideal which is a Maximal additive subgroup is also a Maximal Ideal . any idea how to prove it..
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0answers
36 views

need an example of a ring that is an integral domain but not a field [closed]

We know that every field is an integral domain but every integral domain (has identity, commutative and has no zero-divisor) does not have to be a field. I try to find an example for that, can you ...
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1answer
33 views

Find the number of elements of order 3 in $S_7$

I understand that there are two cycles of length 3, $(i,j,k)(a,b,c) \in$ $S_7$. However, I'm quite stumped in figuring out the logic behind these steps, leading to the answer : Number of distinct 3 ...
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0answers
27 views

Normal Form of Elements in a Group

Suppose that there is a family of groups $A_n$ with $n\in\mathbb{N}$ and $A_1$ is the trivial group. If there is a split exact sequence $$0\to B_n\to A_n\to A_{n-1}\to 0,$$ where structure of the ...
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1answer
29 views

If the nilpotent class of $G$ is $k$, what's the nilpotent class of $G/C_{k-1}(G)$?

I read from a website, it said the nilpotent class will be $k-1$ at most. But why? (I know as the quotient group its class should be $k$ at most.)
2
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1answer
61 views

Yoneda Lemma: definition of Yoneda functors

In stating the Yoneda Lemma, my category theory book (Kashiwara's Categories and Sheaves) makes the following definition: Let $\mathcal{C}$ be a $\mathcal{U}$-category, where $\mathcal{U}$ is ...
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1answer
26 views

$H\leq G$ implies $H^{'}\leq G^{'}$?

Let $G$ be a group. The commutator of $a, b\in G$ is defined as $[a, b]=aba^{-1}b^{-1}$. The commutator subgroup of $G$ is defined as $$G^{'}=\langle [a, b]: a, b\in G\rangle,$$ that is, $G^{'}$ is ...
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1answer
25 views

Stabilizer of an element in the Grassmannian

Let $Gr_{n,k} = \{W\leq \mathbb{C}^n:\mbox{dim(W)}=k \}$ be the set of $k$ dimensional subspaces of $\mathbb{C}^n$. We have a group action: $$GL_n(\mathbb{C})\times Gr_{n,k} \to Gr_{n,k} \ , \ ...
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2answers
64 views

Examples of natural isomorphisms

I am a beginning Category Theory student, and a intermediate Algebra student. Could someone provide me with some examples of natural isomorphisms (in Category Theory) besides the natural isomorphism ...
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0answers
29 views

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime.

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime. Assume $a^m$ has order $n$ and, $m$ and $n$ are not relatively prime. Then ...
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0answers
31 views

Cosets of $H$ in the group $S_4$

In the group $S_4$, let $H$ be the subgroup of those permutations which leave $4$ fixed: $$H=\{(1),(12),(13),(23),(123),(132)\}.$$ List all of the left and right cosets of $H$ in $S_4$.
6
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1answer
70 views

Most elementary proof that a determinant is divisible by $m$

So a challenge problem states that you have an $n \times n$ matrix, where each entry is an integer between $0$ and $9$, and when each row is read as a base-10 number the number is divisible by a ...
1
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1answer
38 views

Solving permutation group equations

Assume we are in $S_7$. Let $\alpha^3=(1,2,3,4)$. How to solve for $\alpha$? This is what I did: $\alpha^3=(1234)$ implies that after $3$ transformations $1 \mapsto 2$. So, begin by letting $\alpha ...
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1answer
22 views

Why does existence of formulas for zeroes of $f(x)=0$ where $f \in K[X]$ (degree $2$ or $3$) imply $f$ is solveable by radicals?

Why does existence of formulae for zeroes of $f(x)=0$ where $f \in K[X]$ (degree $2$ or $3$) imply $f$ is solveable by radicals? It is stated that for quadratic or cubic equations over a field ...
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2answers
31 views

Let $a$ be an element of order $n$ in a group $G$. If $m$ and $n$ are relatively prime, then $a^m$ has order $n$.

Let $a$ be an element of order $n$ in a group $G$. If $m$ and $n$ are relatively prime, then $a^m$ has order $n$. Assume $m$ and $n$ are relatively prime, and that $a^m$ does not have order ...
1
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0answers
36 views

Best way to learn material dealing with cosets, quotient groups and the isomorphism theorems

I'm self studying abstract algebra from Abstract Algebra by Dummit and Foote. I've been able to get through the first few chapters and do problems without any issue, until I hit the material on ...
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1answer
25 views

$R/J(R)$ not semisimple Artinian

I search for a ring $R$ with Jacobson radical $J(R)$ such that $R/J(R)$ is not semisimple Artinian. Being a finitely generated module over itself, $R$ would have infinite hollow dimension due to ...
1
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1answer
16 views

If $G=G_0\geq \ldots\geq G_n=\{1\}$ then $\displaystyle|G|=\prod_{i=0}^{n-1} |G_i/G_{i+1}|$?

Let $G$ be a finite group and $G=G_0\geq \ldots\geq G_n=\{1\}$ a normal series of $G$. How can I show $$|G|=\prod_{i=0}^{n-1} |G_i/G_{i+1}|,$$ that is, the order of $G$ is the product of the orders of ...
0
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1answer
22 views

If $A,B$ are subgroups of the group $G$, then $A\cup B\leq G\Leftrightarrow (A\subseteq B \vee B\subseteq A)$.

Here's the problem and my solution. Please check if the solution to part two is right and give me a hint about the subgroup-ness of $A\cup B\cup C$. Thanks. Problem: If $A,B$ are subgroups of a group ...
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2answers
29 views

An ideal which is not finitely generated

Let $K$ be a field and $A=K[x_1,x_2,x_3,...]$. Prove that the ideal $I:=\langle x_i: i \in \mathbb N\rangle$ is not finitely generated as $A$-module. I have no idea what can I do here, I mean, ...
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1answer
34 views

A question about rings [duplicate]

Suppose $R$ is ring with unit and $a$ is a nonzero element of the ring. If there are two distinct elements $b,c$ such that $ab=ac=1$, show that there are infinite elements $x$ in $R$ such that $ax=1$. ...
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2answers
27 views

abstract algebra , group theory , centralizer

if G is a group with a group element a such that a^5=e , show that c(a)=c(a^3) ? to show that c(a) ⊆ c(a^3) , let g ∈ c(a) then ag=ga . (a^3)g=(a^2)ag=aa(ga)=a(ga)a=(ga)aa=g(a^3) , so , g ∈ c(a^3) ...
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1answer
53 views

Find the centralizer in $S_7$ of $(123)(4567)$.

I am struggling to understand what the centralizer in a permutation of order $7$ means. "The centralizer consists of all elements that commute with $(123)(4567)$" but.. is there a more rigid ...
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1answer
34 views

A question regarding the number of generators of an ideal [duplicate]

Let $I$ be an ideal in $\mathbb{C}[x_1 ,x_2 ,x_3 ,x_4 ]$ such that $I$ is generated by $x_1 x_3$, $x_2 x_3$, $x_1 x_4$, and $x_2 x_4$. How to show that this I cannot be generated by two ...
0
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1answer
29 views

Primitive polynomial and divisibility

Let $f(x) \in \mathbb Z[x]$ with $c(f)=1$ and $f$ is non-constant. Now suppose $h(x) \in\mathbb Z[x]$ be such that $h(x)=f(x)q(x)$ where $q(x) \in\mathbb Q[x]$. Then I have to show that $q(x) ...
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votes
1answer
18 views

Fined the order of the following subgroup of $A_n$

For a positive integer $n\geq 4$ anfd a prime number $p\leq n,$ let $U_{p,n}$ denote the union of all p-sylow subgroups of the alternating group $A_n$ on $n$ letters. Also let $K_{p,n}$ denote the ...
0
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2answers
43 views

Showing that an ideal is maximal

Let $k$ be an algebraically closed field and $f$ be the polynomial $x_1x_2+x_2x_3+x_3x_1$ in $k[x_1, x_2, x_3]$. Here $f$ is irreducible. Then this polynomial ring is not a $PID$, it is only an ...
0
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1answer
48 views

Solving an exercise in Pinter's Abstract Algebra

It is an exercise 5ch,D7 in Pinter's Abstract Algebra. Let $H$ be a subgroup of $G$, and let $K = \{a \in G : axa^{-1} \in H$ for every $x \in H\}$. Prove that $K$ is a subgroup of $G$. I have ...
1
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1answer
26 views

Show $\alpha^m = \varepsilon$ working with permutation groups

Show that $\alpha^m = \varepsilon$ using $\alpha^\ell (a_i) = a_{(i+\ell) \bmod{m}}$ where $\alpha = (a_0 a_1 \dots a_{m-1}) \in S_n$ a permutation group. I've been working on this problem but can't ...
2
votes
1answer
31 views

What are the implications of knowing the algrebaic structure(group, ring, monoid, etc) of a set?

I remember groups, rings, monoids, lattices, etc. being taught in my undergraduate mathematics course. I never really understood what they were for. After that lesson, we moved on to other lessons ...
1
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0answers
27 views

Locally cyclic module exercise

An $A$-module $M$ is locally cyclic if every submodule of $M$ of finite type (finitely generated) is cyclic. (i) Show that every submodule of a locally cyclic module is locally cyclic. (ii) Prove ...
1
vote
1answer
28 views

Unique homomorphism both ways

Suppose that $G_1$ and $G_2$ are groups and that there exists a unique homomorphism $f:G_1\rightarrow G_2$ and a unique homomorphism $h:G_2\rightarrow G_1$. Will it be true then that $G_1$ and $G_2$ ...