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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I don't understand this notation- abelian groups

May be a stupid question but is $(\mathbb{Z}^n)_p \equiv \mathbb{Z}^n/(\mathbb{Z}p)^n$ (when $p$ is a prime)??
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Which of the following statements are true (C.S.I.R - 2014)

Given any psitive integer n, there exist a field extention of $\mathbb Q$ of degree n. Given a positive integer n, there exist a field $F$ and $K$ such that $F \subseteq K$ and $K$ is Galois Over ...
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1answer
17 views

Listing the elements of $U(\mathbb{Z}_{54})$.

The set of all integers modulo $q$ is denoted by $\mathbb{Z}_q$. When equipped with multiplication modulo $q$, has the structure of a commutative monoid, the identity element being equivalence class ...
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1answer
26 views

Integral domain without unity has prime characteristic?

By an integral domain, we mean here, a ring (not necessarily with unity) in which $ab=0$ implies $a=0$ or $b=0$. Question: If an integral domain without unity has positive characteristic, is it ...
4
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1answer
37 views

Divisibility of group exponents when the subgroup has finite index.

Let $G$ be a group (not necessary finite) and $H$ a subgroup of $G$ of index $n$ such that exp $(H)<+\infty$ . Show that $$\exp(G)<+\infty$$ and $$\exp(G)\mid\exp(H)\cdot n.$$ Remarks. ...
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1answer
19 views

Height and coheight of an ideal

Given an ideal $\mathfrak{a}$, Matsumura defined the height of $\mathfrak{a}$ as: $$\text{ht}(\mathfrak{a})=\inf_{\mathfrak{p}\in V(\mathfrak{a})}\text{ht}(\mathfrak{p})$$ He states that: ...
2
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1answer
25 views

tesnsor product of two copies of $\Bbb{R}$ over $\Bbb{R}$

I would like to know what would be tensor product of set of reals over reals would be? That is, $\Bbb{R} \otimes_\Bbb{R} \Bbb{R}$ I think it should be $\Bbb{R}^{2}$ as tensor product combines two ...
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1answer
51 views

Tensor product of (general?) groups

I am starting to learn about tensor products of abelian groups. Why is the tensor product defined for abelian groups? In which part of the construction the commutativity of the groups is needed?
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37 views

Canonical algebra isomorphism $k[D(f)]\cong k[S_0,\dots,S_n]_{(f)}$?

Here's a common set up. Suppose you have $f\in k[S_0,S_1,\dots,S_n]$ is a homogeneous polynomial with $\deg(f)=d$, over some closed field $k$. Let $D(f)$ be the principal open set of $f$ in projective ...
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1answer
126 views

Group theoretic solution to an IMO problem

Is there a (strictly) group theoretic interpretation (and possibly a solution) to this problem (taken from the 27th IMO)? "To each vertex of a regular pentagon an integer is assigned in such a way ...
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2answers
74 views

Group of order 30 can't be simple

I have this following question from my class note on Sylow Theorem: Show that a group of order 30 can not be simple. For that I know the followings: (1) A simple group is one that does not have ...
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1answer
24 views

Prime numbers $p$ and $q$ and possession of normal subgroup of order $p$

I have this following question from my class note on Sylow Theorem: Let $p$ and $q$ be prime numbers such that $p \nmid (q-1).$ Show that each group of order $pq$ possesses a normal subgroup of ...
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60 views

A question about the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$

I wanted to clarify some confusion I was having on the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$, which I call $Aut(\mathbb{Z}_{2} \times \mathbb{Z}_{4})$. I considered the ...
2
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1answer
35 views

Is my proof correct? ($A_n$ is generated by the set of all 3-cycles for $n \geq 3$)

I want to prove that for $n \geq 3$, the alternating group $A_n$ is generated by the set of all 3-cycles. Here is my attempt: Let $\mathcal{S}$ be the set of all 3-cycles in $S_n$, which is a ...
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0answers
29 views

finite index subgroups of profinite completions

Let $G$ be a finitely generated, residually finite group, and let $\widehat{G}$ denote its profinite completion. Is there a 1-1 correspondence between finite index subgroups of $G$ and open subgroups ...
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2answers
303 views

If consecutive elements commute each other, does it mean that all of them commutes with each other?

Let $x_1,x_2,...,x_k$ be $k$ different elements of a group $G$ and $k\geq4$. If we know that $x_i$ commutes with $x_{i+1}$ and $x_k$ commutes with $x_1$, can we say that all $x_i$ commutes with each ...
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1answer
35 views

A non-UFD where prime=irreducible [duplicate]

It is easy to see that in an atomic domain (where every element factors into irreducibles), we have that all irreducibles are prime iff the domain in question is an UFD. I think it is not true for a ...
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33 views

Irreducibility of a polynomial with algebraically independent coefficients

I am learning some kind of field theory. Let $\mathbb{Q}'$ be the smallest subfield in $\mathbb{C}$ containing all roots of unity. Recently I read a book on Galois theory and met the following ...
4
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2answers
67 views

Proof of the properties of tensor product

On page 25 of Atiyah-Macdonald "Introduction to commutative algebra", the author says that "We shall never again need to use the construction of the tensor product given above and the reader may ...
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1answer
38 views

Quick question: G-set functor

The Wikipedia page on Representable Functor says: A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds to a ...
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37 views

In a finite ring, if $xy=1$, show $yx=1$. [duplicate]

Assume $L$ is a finite ring with unit. Suppose for some $x, y \in L$ we have $xy = 1$, then show that $yx = 1.$
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23 views

prime and maximal ideals in $\Bbb Z[x]$ [duplicate]

what are the prime ideals and maximal ideals in $\Bbb Z[x]$? I know $Z[x]$ is UFD but not PID, and (x) is prime but not maximal, and (x,2) is maximal. I wonder what should be the form of all ...
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1answer
53 views

Irreducible and prime elements

In my commutative algebra lecture notes it says: A non-zero element $p$ of a ring $R$ which is not a unit of $R$ is called a prime element if $p=ab$ implies $a$ is a unit or $b$ is a unit. Is this ...
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2answers
46 views

$\Bbb{Z}[X]/\langle f\rangle$ is a finitely generated $\Bbb{Z}$-module

Let $f$ be a monic polynomial in $\Bbb{Z}[X]$. Show that $\Bbb{Z}[X]/〈f〉$ is a finitely generated $\Bbb{Z}$-module. I don't even know how to start. If $g\in\Bbb{Z}[X]/〈f〉$, we are trying to find ...
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1answer
45 views

Proving that if the semigroup (A, *) is a group, then the relation is an equivalence relation.

I'm aware that posting exam questions is probably frowned upon, but this isn't homework, I think I'm genuinely misunderstanding some part of the algebra. The question is this: Throughout this ...
4
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1answer
52 views

Homomorphism between finite groups

I have to prove or disprove the following statement: If $\phi:G \rightarrow H$ is a homomorphism between finite groups, with non-trivial image (i.e. $\phi(G)\neq\{e_H\}$), then $\#G$ and $\#H$ ...
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2answers
44 views

$\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module

I'm trying to show that $\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module. Assume to the contrary that $$\Bbb{Q}=\Bbb{Z}\dfrac{a_1}{b_1}+...+\Bbb{Z}\dfrac{a_n}{b_n}$$ where $a_i,b_i\in\Bbb{Z}$. ...
4
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3answers
59 views

Does there exist a unital ring whose underlying abelian group is $\mathbb{Q}^*$?

Let $\mathbb{Q}^*$ be the group of units of the rational numbers. Does there exist a unital ring whose underlying additive group is $\mathbb{Q}^*$? I don't really have a gut feeling yea or nea. ...
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1answer
27 views

Isomorphism of quotient of direct sum modules

Let $M, N, M'$ and $N'$ be R-modules. If $M'$ and $N'$ are submodules of both $M$ and $N$ then is it true that \begin{equation} \frac{M}{M'} \oplus \frac{N}{N'} \cong \frac{M \oplus N}{M' \oplus N'} ...
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1answer
34 views

Homomorphism from a finitely generated module to a direct sum of modules

Let $R$ be a commutative ring with unit. If $M$ and $N_i$ are arbitrary $R$-modules, the module $\operatorname{Hom}_R(M,\bigoplus_{i\in I}N_i)$ is not isomorphic to $\bigoplus_{i\in ...
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1answer
41 views

judge if nilradical equals jacobson radical

judge if nilradical equals jacobson radical 1)a noetherian ring that is not a artin ring. 2)a local integral domain that is not a field. 3)a integral domain with only finite number of ...
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0answers
53 views

Evaluate a rational function at infinity

In the context of the Tate pairing, I would like to know that it means to `evaluate' an $\mathbb{F}_{q^k}$-rational function at $\infty$. For instance, the reduced Tate pairing is $e_n:G_1\times ...
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0answers
60 views

free group with 2 generators (two matrices)

Let $\alpha$ be complex number such that $| \alpha | > 1$. Show that $\left(\begin{array}{cc}1&0\\\alpha&1\end{array}\right) $ and ...
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1answer
22 views

Showing that Euclidean domain is UFD

Let $D$ be euclidean domain. We claim the following: $1)$ Every element of $D$ can be expressed as a product of irreducible elements $2)$ Every irreducible element of $D$ is a prime element. From ...
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1answer
46 views

Example of non noetherian ring and noetherian $\Bbb Z$-module

a non Noetherian ring that is a Noetherian $\Bbb Z$-module a Noetherian ring that is a non Noetherian $\Bbb Z$-module I have no idea in 1, and I'm not sure if $\mathbf{Q}$ is right for 2? ...
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2answers
36 views

How to prove homomorphism identity

I have problems to do the problem in http://www.math.helsinki.fi/kurssit/alggeom/h1.gif Let $k$ be a commutative ring and $f\in k[X_1,\ldots,X_n]$. Let $k^\prime,k^{\prime\prime}$ be commutative ...
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1answer
48 views

How can I set a Polynomial Algebra?

I was reading about Algebra over a field, and I see the definition of $\mathbb{K}\langle X \rangle$ as follows: Let $X \neq \emptyset$ a set, a word $w$ is an expression in the following way: ...
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1answer
44 views

An Infinite Cyclic Group has Exactly Two Generators: Is My Proof Correct?

I have completed a proof of this that I am inclined to believe is correct, or at least on the right track. I would like to ask if it is indeed correct, or if I need a nudge in the right direction. ...
2
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1answer
33 views

Subgroups and an union of orbits

I have to prove or disprove the following statement: If a group $G$ acts on a set $X$, then every subgroup $H$ of $G$ acts on the set $X$ as well, and every orbit of the action $G$ on $X$ is an ...
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1answer
53 views

Why field of fractions of $k[x_1,x_2,…]$ is Noetherian? [on hold]

the classical counterexample of a subring of a noetherian rings that is not noetherian is $k[x_1,x_2,...]$, which is not noetherian, but the field of fractions of $k[x_1,x_2,...]$ is, can anyone ...
2
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2answers
48 views

Group theory, the squares of G

We have a group $G$ with a subgroup $G_2$, which is defined by $G_2:=\{g^2|g \in G \}$. I have to prove that i) $G_2\triangleleft G$ ii) all elements of $G/G_2$ have order $\leq2$ iii)if ...
4
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2answers
66 views

Free finitely generated modules

Let $A$ be a ring and consider the free modules $A^{\oplus n}$, $A^{\oplus k}$, with $n,k\in \mathbb{N}$. Can $A^{\oplus n}$ be isomorphic to $A^{\oplus k}$ if $k\neq n$? Thanks in advance for the ...
2
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1answer
46 views

Finding a surjective homomorphism

I have to show that for all groups with $2007(=3^2\times223$) elements that there exists a surjective homomorphism to a group of 9 elements. Obviously a group with 2007 elements has a subgroup of ...
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1answer
36 views

if $A$ and $B$ are subnormal, then $A\cap B$ is subnormal [on hold]

A subgroup $X$ of a group $G$ is said to be subnormal if there exists a series $$X=X_0\subseteq X_1\subseteq\cdots\subseteq X_n=G$$ where each $X_i$ is normal in $X_{i+1}$. Prove that if $A$ and $B$ ...
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1answer
36 views

Finite non-abelian group $G$ containing a subgroup $H_0\neq\{e\}$ with property that $H_0\leq H$ for all subgroups $H\neq\{e\}$ [on hold]

Give an example of a finite non-abelian group $G$ containing a subgroup $H_0\neq\{e\}$ with property that $H_0\leq H$ for all subgroups $H\neq\{e\}$ of $G$ This question is from Herstein Topics in ...
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3answers
57 views

Prove that an ideal is not maximal

Ring $\mathbb Z[x],$ ideal is $(x)$. How to prove that this is NOT a maximal ideal? I can't imagine ideal, part of which would be $(x)$.
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49 views

Subgroups of $S_{n}$ of index $n$

We know that for $n\geq{5}$ any subgroup of $S_{n}$ of index $n$ is isomorphic to $S_{n-1}$. We know that by looking at the set of functions that fix a given $j$, we can obtain $n$ such subgroups ...
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1answer
18 views

$|\{ x\in X: g.x=x \space\space\space \forall g\in G \}| = |X|\space mod \space p$

Let $G$ be a p-group. $|G|=p^n$ for some n. Let X be a finite set so that $\,p\nmid |X|\,$, G acts upon X. Denote $A:= \{ x\in X: g.x=x \space\space\space \forall g\in G \}$ I am trying to show ...
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2answers
45 views

every ideal is contained in a maximal ideal

The statement is: In a commutative ring with 1, every proper ideal is contained in a maximal ideal. and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset A\mid A\text{ is ...
6
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1answer
75 views

inverse limit in the plane

What stuff can I say about inverse limits regarding the mapping of $[0, 1]$ onto $[0,1]$ given by $$f(x) = \left\{ \begin{array}{ll} 2x & \mbox{if } 0 \le x \le {1\over2}\\ 1 & \mbox{if ...