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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The ring of upper triangular matrices as a module over itself

$R$ is taken to be the ring of upper $3 \times 3$ matrices with entries in $\mathbb{R}$. If I view $R$ as a module over itself, are any of its submodules free? And how can I prove that its ...
2
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2answers
58 views

From the viewpoint of categorial logic, how should the notion of a topological $R$-module be defined?

Edit. What I really want is to view a topological $R$-module as being a model of some theory $T$ (dependent on the topological ring $R$) in $\mathrm{Top}.$ Can this be done? Original question. From ...
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1answer
125 views

Find all homomorphisms from a quotient polynomial ring $\mathbb{Z}[X] /(15X^2+10X-2)$ to $\mathbb{Z}_7$

I'm completely lost, what my problem is I don't get the gist of a quotient polynomial ring nor ANY homomorphisms between it and some $\mathbb{Z}_n$, much less ALL of them. I know there is something ...
2
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1answer
71 views

Is weak Hilbert Nullstellensatz's theorem “if and only if”?

My question is quite simple, I would like to know if the weak Hilbert Nullstellensatz's theorem can be "if and only if": Nullstellensatz theorem If $I\subset k[X_1,\ldots,X_n]$ is a proper ideal, ...
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1answer
54 views

Does Cartesian Product and Collection of all Sets Perform a Semigroup?

We know that the Cartesian Product is a binary operation. Also it is an associative operation. We know that Cartesian Product of two set is again set, there is even closure axiom. So I need to know ...
6
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3answers
591 views

Integral domain whose irreducible elements are not prime

Is there some integral domain such that none of its irreducible elements is prime? Recall that a nonzero, non invertible element $a$ of an integral domain $D$ is said to be Irreducible, if for ...
0
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1answer
38 views

the pairs which are conjugate to each other in the respective groups.

Pick out the pairs which are conjugate to each other in the respective groups:- (a)$\begin {bmatrix} 1 & 1 \\0&1 \end {bmatrix} $ and $\begin {bmatrix} 1 & 0 \\1&1 \end {bmatrix} ...
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3answers
175 views

Pick out the rings which are integral domains

Pick out the rings which are integral domains: a. $\mathbb{R}[x]$, the ring of all polynomials in one variable with real coeffcients. b. $C^1[0, 1]$, the ring of continuously differentiable ...
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0answers
98 views

What is $Spec(\mathbb{Z}[x])$? [duplicate]

What is $Spec(\mathbb{Z}[x])$? For a commutative ring $A$ e with $1$, its spectrum $Spec(A)$ is defined to be the set of all of its prime ideals. So the question is to find all the prime ideals of the ...
2
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2answers
284 views

Should the sum of zero divisors also a zero divisor?

In a general ring $A$ (commutative with $1$), should the sum of two zero divisors also a zero divisor? Could anyone give a proof or a countexample? Moreover, consider the polynomial ring $A[x]$, ...
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1answer
138 views

Prove that if $A \vartriangleleft G$ is abelian, then $A$ has a complement in $G$.

Let $G$ be a finite group. Suppose that the intersection of all of the maximal subgroups of $G$ is trivial. Prove that if $A \vartriangleleft G$ is abelian, then there exist $U \subseteq G$ such that ...
3
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1answer
215 views

Let $G$ be a finite group. Suppose $N,H,K \subset G$ are subgroups such that $NH=G$ and $(N \cap H)K=G$. Prove that $N(K \cap H)=G$. [closed]

Let $G$ be a finite group. Suppose $N,H,K \subset G$ are subgroups such that $NH=G$ and $(N \cap H)K=G$. Prove that $N(K \cap H)=G$. I have no idea. Give me some hints.
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0answers
83 views

What is a kernel? [duplicate]

It seems the term 'kernel' pops up all over the place and has different meanings everywhere. Is there a unifying feature to all things called 'kernel' in math that would better help me understand what ...
2
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4answers
213 views

Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then $x$ is not integral over $S$

Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then I want to show that $x$ is not integral over $S$ I'm not seeing why this is the ...
0
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1answer
127 views

Show that either $x$ or $1-x$ is invertible in $R$ [closed]

Let $F$ be a field, and let $R$ be a subring of $F$. Suppose that for each $u\in F\setminus \{0\}$, either $u\in R$ or $u^{-1}\in R$. Given $x\in R$, show that either $x$ or $1-x$ is invertible (or ...
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1answer
69 views

Given a ring $R$, when one talks of $R[X]$ what do they exactly mean?

Given a ring $R$, when one talks of $R[X]$ what do they exactly mean? To elaborate, do they mean ring of all elements of the form $a_nX^n + \ldots a_0$ where NO equivalences are present? I.e $a_nX^n ...
0
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4answers
64 views

$A_f$ is not a local ring

I am searching for an example where $A_f$ is not a local ring, where $A$ is a commutative ring with unity, and $f$ is an element of $A$, and $A_f$ is the localized ring with multiplicative set $S=$ ...
0
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1answer
58 views

Subspace of Division Algebra

I'm working on understanding the following proof: https://dl.dropboxusercontent.com/u/17606191/proof.gif but I'm having some trouble understanding some of the author's terminology. We're asked to ...
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1answer
90 views

How to describe free magmas in more structuralist terms?

Given a generating set $G$ (assume for simplicitly it consists entirely of urelements), the free magma on $G$ can be described concretely as follows. Its underlying set is the least $U \supseteq G$ ...
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2answers
75 views

Name for this axiom $(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$

I am trying to give a name to this axiom in a definition: $(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$ (for all $X, Y, R, S$) where $\sqcup$ is the join of a ...
3
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4answers
66 views

$Z^2-YZ-Y^2+X^2+2XY$ is an irreducible polynomial

How to show that $Z^2-YZ-Y^2+X^2+2XY$ is an irreducible polynomial in $\Bbb{C}[X,Y,Z]$?
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1answer
37 views

Integral dependence relation

Let $K$ be a field. Then $K[X^2]$ in contained in $K[X]$, and it is a finite ring extension. Now let $P(X)$ be a polynomial of $K[X]$. What is the polynomial that $P(X)$ satisfies over $K[X^2]$? Can ...
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1answer
27 views

“Maximal Commutative subset”

I'm looking at a proof of Frobenius Theorem, but I can't understand what is meant by the term "Maximal commutative subset." The relevant part of my link is in the proof of (1). The author supposes ...
3
votes
1answer
105 views

A finite group which has exactly eight Sylow $7$-subgroups

Let $G$ be a finite group which has exactly eight Sylow $7$-subgroups. Prove that there exist a normal subgroup $N$ of $G$ such that its index is divisible by $56$ but not by $49$. Give me some ...
2
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3answers
122 views

Number of elements which are cubes/higher powers in a finite field.

This question is a slight generalization of This Question. How many elements are there in a finite field of order $q$ which are : Squares. Cubes. Higher powers. I mean : How many elements are ...
0
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1answer
218 views

Jacobson radical of a finite commutative ring [closed]

Let $R$ be a finite commutative ring, and let $J$ be the Jacobson radical of R (the intersection of all the maximal ideals of R). (1) Prove that $J^n=0$ for some $n$. (2) Use the Chinese ...
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2answers
112 views

Suppose $γ$ is a $k$th root of unity that satisfies a quadratic equation $z^2−mz−n=0$ with $m,n\in\mathbb{Z}$. Then $k=3,4$ or $6$

Let $k\in\mathbb{Z}$ with $k>2$ and suppose $\gamma$ is a $k$th root of unity that satisfies a quadratic equation $z^2-mz-n=0$ with $m,n\in\mathbb{Z}$.Then $k=3,4$ or $6$. My knowledge on ...
2
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2answers
96 views

$X^6 + 3X^4+3X^2-1$ is the minimal polynomial of $\sqrt{ \sqrt[3]{2}-1}$ over $\mathbb Q$

It can easily be seen that $\sqrt{ \sqrt[3]{2}-1}$ is a root of $X^6 + 3X^4+3X^2-1$, which should be its minimal polynomial. Let $a=\sqrt{ \sqrt[3]{2}-1}$. Then $\sqrt[3]{2} = a^2+1$. Therefore ...
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1answer
82 views

Verifying Injectivity of Some Module

So I have the folowing ring $R= \left\lbrace \left(\begin{smallmatrix}a&b&c\\0&d&e\\0&0&f\end{smallmatrix}\right) : \mbox{all entries are in some field } \mathbb{K} ...
0
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2answers
50 views

Semisimple module without composition series

I can't find a semisimple module that doesn't have a composition series. I know every semisimple ring has a composition series as subrings, but i am not sure about modules. Edit: Am right about ...
0
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2answers
126 views

The polynomial $f (t) = t^7 + 10t^2 − 5$ has no roots in the field $\mathbb Q(\sqrt[3] 5, \sqrt[5] 5)$.

Prove that the polynomial $$f (t) = t^7 + 10t^2 − 5$$ has no roots in the field $\mathbb Q(\sqrt[3] 5, \sqrt[5] 5)$.
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3answers
188 views

Why $(\mathbb{Z}, + )$ is not isomorphic to $(\mathbb{R}^+, *)$?

Can someone explain to me why $(\mathbb{Z}, + )$ is not isomorphic to $(\mathbb{R}^+, *)$ where $*$ is multiplication. My book says they aren't really isomorphic and doesn't say why. I thought that ...
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2answers
299 views

Prove that the only homomorphism between a simple non-abelian group G and abelian group A is trivial

Prove that the only homomorphism between a simple non-abelian group $G$ and abelian group $A$ is trivial. OK. So G is a perfect group (G' = G) and A is abelian (A' = {1})
3
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1answer
346 views

What are some algebraically closed fields?

What are some examples of algebraically closed fields? Wikipedia lists exactly two: $\mathbb{C}$ and the (complex) algebraic numbers. EDIT: scrolling to the bottom of the Wikipedia article, they ...
4
votes
2answers
140 views

Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime?

This is in follow-up to this question. Let $R$ be a commutative ring with identity and consider the set $Z \subset R$ of zero divisors. If the ideal $I\subset Z$ is maximal with respect to the ...
0
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1answer
61 views

Ideal Generated by $H\cup K$ [duplicate]

If $H$ and $K$ are ideals of a ring $R$, what are the elements of $\langle H\cup K\rangle$? What I am trying to ask is that how does its element look?
7
votes
4answers
186 views

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian.

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian. I can prove that if $\left|G\right|\leq5 $ then $G$ is abelian. Is it equivalent to this ...
8
votes
2answers
263 views

Can the product of two non invertible elements in a ring be invertible?

Let $A$ be a unitary ring. The question is simply: can the product of two non invertible elements in $A$ be invertible? I proved that the answer is negative if $A$ does not have zero divisors, ...
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Prove that the subgroup of the quotient group is cycling and infinitely generated

$$M = \left\{\,\dfrac{m}{13^n}\biggm| m\in \mathbb{Z}, n\in\mathbb{N} \,\right\}, \quad G = M/\mathbb{Z}$$ Prove that any subgroup $H < G$, $H\neq G$ is cyclic and infinitely generated and that ...
3
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3answers
221 views

What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, ...
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1answer
77 views

Fundamental questions on rings of polynomials.

Put $\mathfrak{E}$ the union of $(0,0)$ and $k\times 1$ in $k^2$ ($k$ an algebraically closed field). Furthermore let $\mathfrak{Z}$ the set of all $f\in k[x,y]$ such that $f(s)=0$ for all ...
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0answers
52 views

Necessary and sufficient condition which tensor product of two central simple algebras is a division ring.

I think this is an interesting and important question that when a tensor product of two central simple $K$-algebra is a division ring. If we know the answer, we can make some counterexamples in ...
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63 views

Decomposition subgroup of Cyclic Galois Extension

My question: Say we have a cyclic Galois extension of degree $n$ over $\mathbb{Q}$. Denote the Galois group as $G$. If $H\leq G$, then does there exist a prime, $q$ in $\mathbb{Z} \subset \mathbb{Q}$ ...
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3answers
101 views

For a finite field $F$ of order $n$, all elements are roots of $x^n - x$

I need to prove two things at $F[X]$ but don't know how, Ill glad to get help... $F$ is a finite field. $|F|=n$. We look at $p(x)=x^n-x\in F[X]$ 1. How we can show that every $c\in F$ is a root of ...
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1answer
87 views

Intersection of a PID and a field

There is an exercise in Bourbaki about the intersection $A = k (x,y) [z] \cap k (z, x + yz)$. These are two subrings of $k (x,y,z)$. The first is PID, the second is a field. Bourbaki requests to prove ...
5
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2answers
223 views

How many elements of order $k$ are in $S_n$?

I need to find how many elements of order $k$ are in $S_n$ (where $k \leq n$). So if $k$ is prime, it's easy: $k$ can't be the $\mathrm{lcm}$ of any integers besides itself and one's (which we're ...
3
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1answer
140 views

to prove that the set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal

im asked to prove that the set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal. i cant even start in the proof , ive just defined my set but cant ...
0
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1answer
38 views

Dimension of a vector space when sum and multiplication changes

If a vector space over the complex numbers has dimension $n$, can we change the definitions of sum and multiplication by complex numbers so that the dimension changes?
2
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1answer
36 views

Periodic Groups of Matrices

Is there any example of a non-finite periodic group of matrices over a field? (I mean of course with regard to the product of matrices). Issai Schur proved that a periodic group of matrices over the ...
0
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1answer
40 views

the direct image of an ideal needs not to be an ideal

I need an example of a ring mapping homo such that the image of an ideal needs not to be an ideal ? I found that the image is an ideal if the mapping was an onto one ! so all we need to find a mapping ...