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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The rationals as a direct summand of the reals

The rationals $\mathbb{Q}$ are an abelian group under addition and thus can be viewed as a $\mathbb{Z}$-module. In particular they are an injective $\mathbb{Z}$-module. The wiki page on injective ...
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34 views

Rings with two elements.

Let $K$ be an assotiative ring with 1 nonzero multiplication. Is it true that if $K$ consists of two elements then $K \cong \mathbb{Z}_2?$ It is clear that second element is $0$ and $1 \cdot 1=1, ...
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1answer
71 views

Are the quaternions obsolete in pure mathematics?

I remember I read an article saying that "The quaternions $\Bbb{H}$ are obsolete in pure mathematics since the theory of vectors has been developed enough, however it is useful in computer science". ...
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34 views

Let G be a finite cyclic group of order n. If d is a positive divisor of n , prove that x^d = e has exactly d distinct solutions in G

well i know that for a group to be cyclic then there must exist an element in G for example we call it g such that $G = \langle g\rangle$ and so $g^0 = e$ and $g^0 = g^n = e$ hence ...
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1answer
50 views

List the elements of $\langle\frac{1}{2}\rangle$ in $(\mathbb{Q},+)$ and in $(\mathbb{Q}^*,\times)$.

List the elements of $\langle\frac{1}{2}\rangle$ in $(\mathbb{Q},+)$ and in $(\mathbb{Q}^*,\times)$. where $\mathbb{Q}^*:=\mathbb{Q}\setminus\{0\}$ My attempt: Well, I know that $\langle ...
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1answer
14 views

Morphism of $k$-algebras between abelian group of $n \times n$ matrices and $m \times m$ matrices

Problem Let $k$ be a field and $f:M_n(k) \to M_m(k)$ be a morphism of $k$-algebras ($n,m \in \mathbb N$). Prove that $n$ divides $m$. I have no idea what to do here, I thought that maybe I should ...
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20 views

Show that there exists a prime ideal $P$ of $R$ such that $ I \subseteq P$ and $P \cap S=\emptyset$ [duplicate]

Show that there exists a prime ideal $P$ of $R$ such that $ I \subseteq P$ and $P \cap S=\emptyset$, if $S$ is multiplicatively closed subset of $R$, $I$ is an ideal of $R$ such that $I \cap S = ...
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1answer
51 views

Showing finite lattices are isomorphic to their sets of ideals.

I'm working out a problem from Gratzer '71. We are to show that any finite lattice $L$ $\simeq$ $I(L)$. My attempt was as follow: prove the contrapositive. So assume $L$ is not isomorphic to $I(L)$. ...
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1answer
45 views

A question about Klaus Hulek algebraic geometry

I'm reading Klaus Hulek's algebraic geoemtry and there is something that I can't understand. Here it says that if {p,q} is a counterexample with minimum max{deg p , deg q}, then it can be assumed ...
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84 views

Can $\mathbb{Z}$ be endowed with operations that give it the structure of a field?

Does there exist some definition of addition and multiplication for which the set of all integers is a field?
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1answer
34 views

Find a group $G$ and $H\subseteq G$ that shows $(H\leq G$ iff $ab\in H)$ is not valid if $G$ is infinite.

The theorem says, let $G$ be a finite group with $H\subseteq G$, with $H\neq \varnothing$. $H \leq G$ iff $ab \in H$ for all $a, b \in H.$ I have no idea how to start.
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1answer
31 views

Definition of Tensor Product of Modules

I am really struggling to understand several parts of the definition of tensor product given in my lecture notes: Definition of the tensor product *Denote by L the free A-module with a basis ...
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1answer
40 views

Uniqueness of Duals in a Monoidal Category

Given a monoidal category ${\cal C}$, and $X \in {\cal C}$, we define a left dual of $X$ to be an object $X^*$ such that there exist morphisms $\epsilon:X^* \otimes X \to I$, and $\eta:I \to X \otimes ...
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1answer
65 views

Prove that the kernel of a group homomorphism $\phi$ is a subgroup and that $\phi$ is injective

I am solving the following exercise: Let $\phi : G_1 \rightarrow G_2$ be a homomorphism (where $G_1$ and $G_2$ are groups) and $\ker \phi := \{ g \in G_1 \mid \phi(g) = e \}$ now I have to ...
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1answer
57 views

Give an example of a maximal ideal in a noncommutative ring which is not prime

While trying to find an example, I came up with this: Since if $J$ is an ideal of a the ring $M_n(R)$, where $R$ is a commutative ring, then $J=M_n(I)$ for some ideal $I$ of $R$. IF I could show that ...
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1answer
57 views

Question Regarding a Group G

Let $G$ be a group and let $a,b \ \epsilon \ G.$ Show that $(a * b) * (a' * b') = e$ if and only if $(a * b)$ = $(b * a)$ Note that * is a binary operation, $a'$ and $b'$ are inverses of $a$ and ...
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24 views

Determining if an extension is a normal extension [MORANDI'S BOOK]

I'm dealing with the following problem: Let $K$ be a field and suppose that $\sigma\in\mbox{Aut}(K)$ has infinite order. Let $F$ be the fixed field of $\sigma$. If $K/F$ is algebraic, show that ...
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2answers
30 views

Homomorphism $\phi: \mathbb{Z} \times \mathbb{Z} \rightarrow G$ proof

Let $G$ be a group. Let $h,k \in G$ and let $\phi:\mathbb{Z}\times \mathbb{Z}\rightarrow G$ be defined by $\phi(m,n)=h^mk^n$. Give a necessary and sufficient condition, involving $h$ and $k$, for ...
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1answer
34 views

How to find out if two cycles are conjugate?

Let for example $a=(14395)(26)(78)$ and $b = (154)(2368)(79)$ be elements of $S_9$. I know that by definition, conjugate elements of a group $G$ are elements $x,y \in G$ such that $x=aya^{-1}$ for ...
2
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2answers
47 views

Product of finite order elements in a group

Let $G$ be a group. Let $a,b\in G$ be of finite order. Prove or disprove: (1) If $ab$ has finite order, then $ba$ has finite order. (2) If $ab$ has finite order, then $a^{-1}b^{-1}$ has finite ...
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95 views

Commutator of a group

A commutator in a group $G$ is an element of the form $ghg^{-1}h^{-1}$ for some $g,h\in G$. Let $G$ be a group and $H\leq G$ a subgroup that contains every commutator. $(a)$ Prove that $H$ is a ...
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39 views

What's the best way to define $\mathbb{H}$?

I had once defined $\mathbb{C}$ as $\mathbb{R}\times\mathbb{R}$ as a set, then I found it extremely uncomfortable when doing complex analysis or measure theory. So I changed its definition so that ...
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1answer
42 views

Suppose that $K$ is a field and that $f$ and $g$ are relatively prime in $K[x]$. Show that $f - Yg$ is irreducible in $K(y)[x]$.

I'm a bit confused of the notation $K(y)[x]$, is that simply $K[y][x]$ so... $K[y,x]?$ Anyways, here's my attempt at trying this before I get stuck. Since $f$ and $g$ are relatively prime, that ...
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1answer
51 views

Zerodivisors in polynomial rings over a non-commutative ring

Prove that if $f \in R[x]$ is a zero divisor then $\exists r(\neq 0) \in R$ s.t $rf=0$, where $R$ is a ring. I know that for $(a_0+a_1x+ \cdots ...
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1answer
28 views

Finite Abelian Group Proof

Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ for some prime $p$. I'm not sure what to do. Any proofs or hints ...
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19 views

An alternative proof for the units of $U_q(\mathfrak{sl}_2)$ using Ore extensions.

I would like to establish what the set of units are in the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$. First, I recall the definition of the quantized enveloping algebra- throughout the ...
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151 views

Calculating the Order of An Element in A Group

First of all, I am very new to group theory. The order of an element $g$ of a group $G$ is the smallest positive integer $n: g^n=e$, the identity element. I understand how to find the order of an ...
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1answer
36 views

Is $\alpha$ conjugate to $\beta$?

Let $\alpha=\left(\begin{array}{ccccc} 1&2&3&4&5\\ 2&1&4&5&3 \end{array}\right)$ then is $\alpha$ conjugate to $\beta=\left(\begin{array}{ccccc} ...
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Let $L \supset K$ be an extension field. Prove $K(a_0, a_1, \ldots, a_n) = K(a_0, a_1, \ldots, a_{n-1})(a_n)$ ? Why smallest subfield with properties?

Suppose $L \supset K$ is an extension field. How do one prove rigorously that $K(a_0, a_1, \ldots, a_n) = K(a_0, a_1, \ldots, a_{n-1})(a_n)$ ? I understand that $K(a_0, a_1, \ldots, a_n)$ is the ...
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How can we find an element of largest order in $S_n$ in general? [duplicate]

How can we find an element of largest order in $S_n$ in general? For the small orders we can find by trial and error method. For example in $S_3$: 6 is the largest possible order. Similarly for ...
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1answer
21 views

What is the definition of finitely generated?

To be specific, here is an example. Note that $(\mathbb{Z},+)$ is a group. Definition 1. (Lang) Let $G$ be a group and $S\subset G$. If $G$ is the intersection of all subgroups $H$ of ...
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What would be an example that rank of a subgroup of a free group is greater than the rank of free group?

Let $G$ be a free group and $H$ be a subgroup of $G$. Then, $H$ is also a free group. Let $R_G,R_H$ be ranks of $G,H$ respectively. what would be an example that $R_H>R_G$?
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Divisibility of polynomials in a subfield of a field.

I am trying to prove the following assertion: Let $K\subset L$ be fields, let $f,g\in K[x]$ be such that $f\mid g $ in $L[x]$, then $f\mid g$ in $K[x]$. We clearly have that $fh=g$ for some ...
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Let $H\subset G, K\subset H$ be normal subgroups such that $K$ is normal in $G$ .Describe $(G/K)/(H/K).$

First of all I would like to say hello to all of you! I am new here! :-) I am very pleased to have found this site. So I can help others if they need me and get help when I need it, really a great ...
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24 views

Definition of direct sum of modules?

I have just started studying modules, and am trying to figure out the definition of the direct sum of modules but I'm having trouble since different sources seem to give different definitions, for ...
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142 views

How to show that there is a bijective correspondence between two sets of prime ideals

I'm trying to solve this Algebra Problem, and I'm not quite sure, if I'm on the right way. Let $R$ be a commutative ring and $S \subset R$ a multiplicative subset. Show that $p \to pS^{-1}R$ ...
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3answers
106 views

Is every free subgroup a direct summand?

Let $G$ be an abelian group. Suppose that $F$ is a subgroup of $G$ such that $F$ is free. Does there necessarily exist a subgroup $H\subset G$ such that $G\cong F\oplus H$? Motivation: In Lang's ...
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Is every symmetric group generated by some cycles?

Actually, I have two questions here: How to show that $D_3$, the dihedral group, is isomorphic to $S_3$ in details? Is every symmetric group generated by some cycles? For question (1), I know ...
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1answer
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Group Extension Example

I am trying to find two distinct group extension of $Z_2$ by $Z_3$. One "natural" extension that I found was $(Z_3,Z_2)$. That is $0 \rightarrow Z_3\rightarrow (Z_3,Z_2)\rightarrow Z_2\rightarrow 0$. ...
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Prove that $M$ is a complex.

Let $f:(A,d) \rightarrow (A^{'},d^{'})$ be a chain map. For each $n$ define $$M_{n}=A_{n-1} \oplus A^{'}_n$$ and $\Delta_{n} :M_{n} \rightarrow M_{n-1}$ by $$\Delta_{n}:(a_{n-1},a_{n}^{'}) ...
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What is maximal possible order of an element in $S_{10}$ ? Why?

What is maximal possible order of an element in $S_{10}$? Why? Give an example of such an element. How many elements will there be in $S_{10}$ of that order?
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Splitting in Short exact sequence

I am trying to find whether $\{1\}\longrightarrow\mathbb{Z}\longrightarrow\mathbb{R}\longrightarrow\mathbb{R}/\mathbb{Z}\longrightarrow \{1\}$ splits. My conjecture is it is not as we cannot find a ...
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1answer
31 views

Localizing at maximal ideals and the product

Let $D$ be an integral domain, $M_{i}$, $i = 1,...,r$ be some of its mutually distinct maximal ideals, and $e_{i}$be positive integers for all $i$. Is it true in general that the extension of the ...
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1answer
22 views

the order of a quotient group element

I'm looking at the solution to this question and there's one part that doesn't make sense to me. Let $G$ be a group and $N$ a normal subgroup of $G$. Prove that the order of the element $gN$ in ...
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1answer
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Prove that : A group $G$ of order $p^n$ has normal subgroup of order $p^k$ , for all $0≤k≤n$

Prove that : A group $G$ of order $p^n$ has normal subgroup of order $p^k$ , for all $0≤k≤n$. I already appreciate your hints/answers
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2answers
50 views

Prove a given group is free

From Hungerford's "Algebra": What type of tools does one have to tackle a problem like this? I seem at a loss at how to show a group is free at all. One can consider each group as the homomorphic ...
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1answer
42 views

Tensor analog of Matrix Product

Given two $n \times n$ matrices $A$ and $B$, we can form their matrix product in the usual way. Is there a similar product for tensors? E.g., if one is given two $n \times n \times n$ tensors ...
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Why the Endomorphism ring of module is a proper class of that of abelian group?

Wikipedia: In the category of R modules the endomorphism ring of an R-module M will only use the R module homomorphisms, which are typically a proper subset of the abelian group homomorphisms. I ...
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89 views

Hilbert's Basis Theorem - Clever Proof?

So I am studying commutative algebra at the moment and I have come across the proof of the Hilbert Basis Theorem (the proof I have is the same as the one in Reid's "Undergraduate Commutative ...