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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46 views

Are “noncommutative resolutions” a thing?

I am interested in looking at "resolutions" of modules in noncommutative rings, making some obvious necessary modifications to the definition of a resolution. However, whenever I try to search the ...
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49 views

Why is the product of two negative numbers not negative? [duplicate]

A positive number multiplied by a positive number is positive, but negative multiplied by negative is not negative but it becomes positive. How ?
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1answer
28 views

Ring isomorphism $\Bbb Q[x]/(f)\cong \{c_0+c_1\alpha + c_2\alpha^2:c_i\in \Bbb R\}$

Let $f=x^3+x^2-2x-1\in \Bbb Q[x]$. Let $\alpha\in \Bbb R$ be a zero of $f$. $\Bbb Q[x]/(f)$ is isomorphic to the subring $R=\{c_0+c_1\alpha + c_2\alpha^2:c_i\in \Bbb R\}$ of $\Bbb R$. The map $\...
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24 views

Irreducible polynomial - Artin-Schreier exercise

Can somebody help me with this exercise? Is $f(x)=50x^6 + 6x^5 -10x^4 + 15x^3 + 4x -7$ irreducible in $\mathbb Z[x]$? I know I have to project in $\mathbb Z_5$ and it becomes an artin ...
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25 views

Uniqueness of generator of principal ideal in K[x_1,x_2,…,x_n]

In $K[x]$ (where $K$ is a field), I know that every ideal can be written as $(f)$ for some $f \in K[x]$. Furthermore, $f$ is unique up to multiplication by a nonzero constant in $K$. Is there a ...
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37 views

Is the lattice of normal subobjects of algebraic theories always modular?

The lattice of (sided) ideals of a ring is always modular, basically by the distributivity of the powerset lattice. I know this argument also works for norma subgroups. I was wondering whether this ...
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2answers
29 views

Extension of a finite field to a finite non commutative ring

Can a finite field be extended to non-commutative finite rings so that not all elements of the field commutes with the elements of the ring? I have been trying this taking the examples of matrices.
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28 views

When is showing any elements in a subset of a group is a sufficient condition for a subgroup. [on hold]

Let G/N be an abelian factor group. To show that H is a subgroup of N, it suffices to show that H is a subset of N. Why is it not necessary in this case to demonstrate that the group axiom holds or ...
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1answer
39 views

Prime ideals in a quotient of a DVR

Suppose $R$ is a DVR. So $R$ has two prime ideals - $(0)$ and $(p)$ ($p$ the uniformizer of the maximal ideal). All other ideals in $R$ are powers of $(p)$, i.e. of the form $(p^k), k\geq 2$. I'm ...
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1answer
41 views

Circular definition in slice category?

I am reading Aluffi (Algebra Chapter 0) there he introduces the slice category in a kind of excercise: When thinking about it I got confused about the "nature" of the $Z$ (and $A$). Since they are ...
6
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23 views

Relation between semiring of sets and semiring in abstract algebra.

Let a $\mathcal R$ be a family of subsets in $\Omega$ that is closed under finite union and relative complement. We say that $\mathcal R$ is a ring of sets in $\Omega$. Symbolically, for any $A,B\in\...
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2answers
78 views

Examples of groups that are not subgroup [on hold]

Sorry, my English is poor, but I have a question. It is usual to find the definition of subgroup as: "We define a subgroup $H$ of a group $G$ to be a nonempty subset $H$ of $G$ such that when the ...
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1answer
18 views

$\mathbb{F}$-subalgebra generated by a set

Assume that $A$ is an $\mathbb{F}$-algebra, where by $\mathbb{F}$ I just denote an arbitrary field. Furthermore, if $X \subset A$ is a proper subset, how do we define the $\mathbb{F}$-subalgebra of $A$...
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26 views

Using the Tensor Product Construction to Show Linear Independence of the Standard Basis

For simplicity's sake, let's assume $V = \mathbb{R}^2$ with standard basis $e_1, e_2$. I construct the tensor product $V \otimes V$ as the quotient of the the free vector space $F(V \times V)$ by the ...
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1answer
29 views

Ideal and subring of $End_K(V)$

Let $V$ be an infinite-dimensional $K$-vector space, $E = End_K(V)$ and $I = \{f \in E;\,\,dim(f(V)) < \infty\}$. I want to prove that $I$ is an ideal of $E$ and $Kid_V + I$ is a subring of $E$. ...
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3answers
78 views

I don't understand how this proof works.

My book proves "if $a(x)$ has degree $n$, it has at most $n$ roots." I will just copy the proof here. Proof: If $a(x)$ had $n+1$ roots $c_1, \dots,c_{n+1}$, then by Theorem 2, $(x-c_1) \dots (x-c_{n+...
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2answers
32 views

Basis of a free module consists of exactly n elements

I' ve found the following theorem while preparing for an Algebra exam: Let $R$ be the ring $\Bbb Z/p^k$ with a prime ideal $p$ and $k\ge2$, $M$ a free $R$-module of rank $n$ (i.e. $M\cong R^n$). ...
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0answers
42 views

Extending our number system to include infinities [closed]

Instead of writing infinity using the infinity symbol, could we write such numbers as: |$\mathbb Z$| (size of the set of integer numbers) |$\mathbb R$| (size of the set of real numbers) Then ...
2
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3answers
70 views

If $H$ is a subgroup of $G$ of index $n$, then $g^{n!} \in H \ \forall g \in G$

Let $G$ be a finite group and $H$ a subgroup of $G$ of index $n$, i.e., $[G:H]=n$. Prove that $$\forall g \in G,\; g^{n!} \in H.$$ This is a question I've had in a past exam for Group Theory and I'm ...
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2answers
24 views

About the definitions of direct product and direct sum of modules.

Direct product of modules can contain infinitely nonzero elements, but direct sum of modules must contain finitely nonzero elements. What's the point of the definitions? Why the definitions are ...
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43 views

Attempt to represent gaussian integers with matrices over ${\mathbb Z_+}^{4\times4}$

Let us first consider the generating element for $C_2$ : $$M_1 = \left[\begin{array}{cc}0&1\\1&0\end{array}\right], \text{ and } P_1 = ({M_1})^2 = I_2 = \left[\begin{array}{cc}1&0\\0&1\...
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0answers
26 views

Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups

Let $L\subset \mathbb{Z}$ be the subgroup of $\mathbb{Z}^3$ generated by the elements $(-1,-1,4),(2,4,0),(3,3,8)$. Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups. I've tried creating a ...
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58 views

Prime ideal of a polynomial ring in 6 variables

Let $k$ be a field and $k[x_1,x_2,x_3,y_1,y_2,y_3]$ a polynomial ring in 6 variables over $k$. How to prove that the ideal $(x_1y_2-x_2y_1,x_2y_3-x_3y_2,x_3y_1-x_1y_3)$ is prime in $k[x_1,x_2,x_3,y_1,...
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24 views

Separable but not reduced? [duplicate]

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every field extension $L/\Bbbk$, and reduced if its underlying ring is reduced. Separable always implied reduced, and I found ...
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2answers
166 views

About an article regarding free groups

I am currently reading this paper https://blms.oxfordjournals.org/content/35/5/624.abstract and I have some difficulties to understand two steps of the proof of the main theorem. Let $G$ be a non ...
2
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1answer
24 views

Hopf gradings on complex commutative group rings

Let $G$ be a finite abelian group. The complex group ring $\mathbb{C}G$ admits a structure of Hopf algebra when the multiplication is the usual multiplication in a group ring and the co-multiplication ...
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33 views

Finite dimensional separable algebra is étale

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every extension $L/\Bbbk$. Say it's étale if there's an extension $L/\Bbbk$ such that $L\otimes_\Bbbk A\cong \prod_1^nL$. Here'...
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40 views

Valuation of Index of polynomial with Newton Polygon

I read here (page 237) that the valuation of the index of a polynomial is equal to the number of integer points below its Newton polygon. I am confused how this makes sense--the cited paper (this) ...
2
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1answer
69 views

$A$ a subset of a finite group $G$ with strictly more than $|G|/2$ elements. Show $AA=G$. [closed]

The question asks (a) Let $A$ be a subset of finite group $G$ with strictly greater than $|G|/2$ elements. Show $AA=G$ and (b) Show this can fail in a monoid. I've been working on this for awhile ...
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27 views

Questions Concerning Proof of Artin-Rees Lemma

I have two questions about the proof of the Artin-Rees Lemma presented here: $\textit{Question 1:}$ Am I correct in assuming $I^0=R$? This is the only way some of the statements make sense I think. ...
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67 views

An elementary proof that $k[x,y]/(xy-1)\cong k[x]_x$, where $k$ is a field

Letting $\phi:k[x,y]\to k[x]_x$, $\phi(x)=x$, $\phi(y)=\frac{1}{x}$, we see that $\ker \phi$ is prime, and $(1-xy)\subseteq\ker\phi$. Now, given that $k[x,y]$ has Krull dimension 2, $\ker\phi\neq (1-...
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1answer
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+100

Why does this ring have rank $k!$?

Let $R$ be any ring free of rank $k$ over $\mathbb{Z}$ having non-zero discriminant. Let $R^{\otimes k} = R \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} R$. Then $R^{\otimes k}$ is a ring of rank ...
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79 views

A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic

Let $G$ be a finite abelian group s.t. it contains a subgroup $H_{0} \neq (e)$ which lies in every subgroup $H \neq (e) $. Prove that $G$ must be cyclic. Also what can be said about $o(G)$ ? I'm ...
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1answer
55 views

Need help in understanding the example from Dummit & Foote text

This is an example from Dummit & Foote text. Let $D_{2n}=<r,s:r^n=s^2=1,s^{-1}rs=r^{-1}>$.Since [$r,s$]$=$$r^{-2}$,we have $\langle r^{-2}\rangle=\langle r^2\rangle \le D'_{2n}$....
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169 views

Why would I learn modern category theory if my interest mainly is structured sets, what would I have to gain? [closed]

A long time ago I studied mathematics at the University of Stockholm. I had a romantic view of modern algebra and manage to make the first two algebra courses by self studies in order to immediately ...
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1answer
30 views

Confusion about generated subrings and subfields

In Milne's field theory notes he defines, given an extension $E/F$ and a subset $S\subset E$, the subfield of $E$ generated by $F$ and $S$ as the smallest subfield of $E$ containing both $F$ and $S$. ...
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39 views

Advantages and disadvantages of a particular definition of rings and subrings

My professor defines a ring as a triple $(A, +,\ \cdot)$ such that $(A, +, 0)$ is an abelian group, $(A,\ \cdot)$ is a semigroup and $\cdot$ distributes over $+$. Subsequently, $B\subseteq A$ is said ...
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54 views

Does the uniqueness of splitting fields up to non-canonical isomorphism fall from algebraic geometry?

Does the uniqueness of splitting fields up to isomorphism over the base field fall out from some deep results in algebraic geometry, or is it something special to fields which I shouldn't expect to ...
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20 views

$f:A\to B$ flat homomorphism of rings, $Q$ is a prime ideal of $B,$ and $P=Q^c.$ Why is $B_Q$ a local ring of $B_P?$

I have a question about Exercise 2.18 on Introduction to Commutative Algebra by Atiyah and MacDonald. Let $f:A\to B$ be a flat homomorphism of rings, let $Q$ be a prime ideal of $B$ and let $P=Q^c,$ ...
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1answer
73 views

Convert $x \not\equiv 0$ mod $pq$ to a modulo polynomial [on hold]

$p,q \in \mathbb{P}$, primes For $x \not\equiv 0 \bmod p$ you can write $(x-1)(x-2) \dots (x - (p-1)) \equiv 0$ mod $p$ Is there a way to do the same for a a composite modulus $pq$? Note: $(x-1)(x-...
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48 views

Is it Noetherian and can you find an ideal which is non-finitely generated in it?

I have a lot of confusion in polynomial ring. Please help me explain it. Firstly, as we know, if $R$ is a ring then $R$ is sub-ring of $R[X]$ and $R[X]$ is a sub-ring of $R[[X]]$. I just wanna ask ...
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1answer
25 views

the example of prime near-ring? [closed]

i need your help to find the example of prime near-ring, please. because i'm so confused to find the example of prime near-ring (the ring is near-ring, not a ring). I t
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1answer
50 views

Understand a part of the proof about permutations in a symmetric group on $n$ elements

Let $\sigma$ be an even permutation in $S_n$($\sigma \in A_n$). Assume $\sigma = \tau\sigma\tau^{-1}$ for some $\tau \in S_n$ and assume that the type of $\sigma$ consists of distinct odd integers. ...
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1answer
27 views

How to find a function/operator that satisfies the following conditions

I'm looking for a function that satisfies : 1) Symmetric: $f(x,y) = f(y,x)$ 2) Associative: $f(f(x,y), z) = f(x,f(y,z))$ 3) $f(x,x) = 0$ 4) it would be nice if $f(x,0) = x$, or at least that $g(x) ...
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0answers
17 views

$F(a_1,\ldots,a_n) = F(a_1+t_2 a_n+\cdots+t_n a_n)$ if Extension is Simple?

Suppose that F is an infinite field and that $L = F(a_1,\ldots,a_n)$ is a simple extension of $F$. Can you show, without the assumption that $[L:F] < \infty$ that $\exists t_2,\ldots,t_n \in F$ ...
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0answers
12 views

S is a subgroup of S(n) such that X(n)=n,X belongs to S(n). then show that S is isomorphic to S(n-1). [closed]

Pls help me with define a isomorphic function from S to s(n-1)
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1answer
21 views

If $G$ is a finitely presented group then is the commutator of $G$ isomorphic to the commutator of $F$ mod the relations?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. Let $F$ be the free group with generating set $S$. Let $[F,F]$ and $[G,G]$ be the commutator subgroups of $F$ and $G$ respectively. Let ...
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0answers
23 views

find a 2 sylow subgroup of $G=S_4\times S_3$ and a $3$ sylow subgroup. [closed]

how to find a $2$ sylow subgroup of $G=S_4\times S_3$ and a $3$ sylow subgroup. and the number of each number of subgroup.
1
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1answer
35 views

Related to multiplicative subgroup of positive real line

Let $F$ be a subgroup of the multiplicative group $\mathbb R^*_{>0}$ such that $F$ is dense in $\mathbb R^*_{>0}$, $$N\cap F=\emptyset\ \text{ and }\ NF=N,$$ in which $N$ is a subset of $\...
0
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0answers
27 views

Why $a+b$ is a generator of $F(a,b)$ over $F$, where $F$ is a field of characteristic zero.

Let $F$ be a field of characteristic zero. Assume that $a$ and $b$ are algebraic over $F$. The primitive element theorem says that there exists $w \in F(a,b)$ such that $F(a,b)=F(w)$; such $w$ is ...