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$\operatorname{Hom}_k(R,k)$ is injective indecomposable $R$-module for $R$ local $k$-algebra of finite $k$-dimension

I am working on Exercise 3.1.22. from Bruns and Herzog's Cohen-Macaulay rings. The exercise in question is the following (the references I will use are of course from within the book): It appears ...
Joel Castillo Rey's user avatar
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A subgroup with finitely many conjugates which has finite index in its normal closure is finite

I was wondering if the following statement is true: let $G$ be a group and $H$ a subgroup of $G$ such that $H$ is almost normal in $G$ (i.e. $H$ has finitely many conjugates in $G$ or equivalently the ...
L.L's user avatar
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2 answers
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Does the monoid of non-zero representations with the tensor product admit unique factorization?

Let $(M, \cdot, 1)$ be a monoid. We will now define the notion of unique factorization monoid. A non-invertible element in $M$ is called irreducible if it cannot be written as the product of two other ...
Smiley1000's user avatar
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Is the direct product of finitely-generated groups cancellative? [duplicate]

The direct product is cancellative for finite groups, so I wanted to know if this result holds for finitely-generated groups as well. The proof linked clearly doesn't apply there, but I have been ...
Zoe Allen's user avatar
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Clarification on Field Homomorphisms

Let $F, k$ be ordered fields. It is clear that a homomorphism $\phi : F \to k$ satisfies the properties of a ring homomorphisms, that is, preserving operations and multiplicative identity. But is it ...
n1lp0tence's user avatar
3 votes
1 answer
83 views

How to determine $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4,\Gamma_5$?

I started learning about Dirichlet Characters. Here is what I learned so far: Definition: Let $m \in \mathbb{N}$. We call a function $\chi:\mathbb{Z} \rightarrow \mathbb{C}$ a Dirichlet Character mod ...
NTc5's user avatar
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1 vote
1 answer
61 views

Direct sum of free abelian group and quotient of abelian group by subgroup

I'm currently studying abelian groups in Kurosh's The Theory of Groups. I'm trying to understand the proof of the theorem: Let $B \leqq A$ be abelian groups. If $A/B \cong C$ and $C$ is a free group, ...
MathematicallyUnsound's user avatar
4 votes
1 answer
54 views

Is the localization of a zero-dimensional ring a quotient?

If $R$ is any commutative zero-dimensional ring and $m$ is a maximal ideal, then is $R_m$ always naturally a quotient of $R$? In other words, is the natural map $R\to R_m$ always surjective? I was ...
Anon's user avatar
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34 views

Profinite completion of profinite groups

I was trying to prove that $\mathbb{R}/\mathbb{Z}$ cannot be a galoisgroup for any extension. My plan for this was to show that $\mathbb{R}$ is a divisible group and that every quotient of a divisible ...
potenzenpaul's user avatar
4 votes
1 answer
71 views

Attaching an element to a ring $R = \mathbb{Z}/(p^{k}\mathbb{Z})$, assuming it is not in $R$

Let $R = \mathbb{Z}/(p^{k}\mathbb{Z})$, where $p$ be any prime number, and $k > 1$ be any integer. Now let us consider an equation $x^r = p$ in $R$ and $\pi$ be the root of this equation, where $r \...
Afntu's user avatar
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Alternative solution to showing that $\langle x^2 +1, y\rangle$ is a maximal ideal and its possible generalization?

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg, and the following Notes: $\langle x^2 +1, y\rangle$ is maximal, pg.3 Question (5a) Background Notation 1: $\...
Seth's user avatar
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What do the words "descends" and "Induced" mean in the following quoted passage?

The following is taken from pg 4 section 6.1 of the following notes Background $\quad$ We start by observing that a ring homomorphism descends to quotient rings whenever the image of an ideal on the ...
Seth's user avatar
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Galois group of $\mathbb{C}(t)$ over $\mathbb{C}(t+t^{-1})$

I was asked to determine the Galois group for the following extensions: first $\mathbb{C}(t+t^{-1})\subset \mathbb{C}(t)$ and then $\mathbb{C}(t^n+t^{-n})\subset \mathbb{C}(t)$ for a certain $n \in \...
riescharlison's user avatar
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1 answer
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$Out(F_n)$ has a free abelian subgroup of rank $2n-3$

Proposition 9.5.4 The group $Out(F_n)$ has a free abelian subgroup of rank $2n-3$. This is a proposition of the book "Topological Methotods in Group Theory" by R. Geoghegan. In the proof he ...
Greg's user avatar
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1 answer
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Factoring polynomials over rings and fields

I would like to determine the factors of $X^N - 1$ over different integer rings, such as $$ \mathbb{Z}[X]\ \mbox{and}\ \mathbb{Z}_p[X]\quad \mbox{for}\quad prime\,\, p $$ In particular, I am ...
user64060's user avatar
3 votes
0 answers
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Are there counterexamples of "dividing each other implies association" on a commutative ring but not integral domain? [duplicate]

I am reading about "Fraction on Commutative Ring". On the textbook a proposition states that Let $R$ be a domain and $a,b\in R$. If $a\mid b$ and $b\mid a$, then there exists a unit $u$ ...
Luost2r's user avatar
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0 answers
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What is wrong in this proof and where is it needed that $R$ is an integral doamin? [duplicate]

I wanted to prove the following theorem myself: Let $R$ be an integral domain and $p \in R$. If $(p)$ is a maximal ideal, Then $p$ is a prime element. My attempt: Since $(p)$ is maximal then there ...
Physor's user avatar
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When does the eliminant (resultant) from three variables into two variables vanish?

Definition. Consider a ring $\mathcal{R}$ and polynomials $P,Q\in \mathcal{R}[x]$. We define the eliminant $\mathrm{Elm}(P,Q)$ of $P$ and $Q$ by the determinant of their Sylvester matrix. If $P(x)=\...
Derso's user avatar
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2 votes
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Comparing mathematical objects by the "rigidity" of their definitions

A loose interest of mine recently has been ordering mathematical objects by how "combinatorial" their study is, in broad terms. I consider the study of a mathematical object more “...
safsom's user avatar
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1 answer
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$\operatorname{End}_S(M \otimes_R S) \cong \operatorname{End}_R(M) \otimes_R S$?

Let $R$ and $S$ be commutative rings with unity. Let $f: R \to S$ be a ring homomorphism. Let $M$ be an $R$-module. Do we have $$ \operatorname{End}_S(M \otimes_R S) \cong \operatorname{End}_R(M) \...
Smiley1000's user avatar
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2 votes
2 answers
92 views

Irreducible factors of $X^n -1$ in $\mathbb{F}_q[X]$

I am stuck trying to prove the following corollary: Let $f = X^n -1 \in \mathbb{F}_q[X]$ with $gcd(q,n)=1$. Let k = order of q mod n. Then the degree of every irreducible factor of $f$ divides k. It ...
Very Interesting's user avatar
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0 answers
27 views

Algebraic simplification of nested radicals and fractions in general?

Let ${}^{-}$ denote the algebraic closure, $n\in\mathbb{N}_+$, $\mathbb{K}\in\{\mathbb{Q},\mathbb{C}\}$, $A(x_1,...,x_n)\in\overline{\mathbb{K}(x_1,...,x_n)}$ (means a rational or irrational $\mathbb{...
IV_'s user avatar
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0 answers
38 views

Correspondence theorem and Quotient ring isomorphism

The following is taken from Abstract Algebra A Comprehensive Introduction by: Lawerence and Zorzitto. Background The ideal in a commutative ring $R$ generated by eleements $a_1,\dots, a_n$ is denoted ...
Seth's user avatar
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1 vote
1 answer
50 views

...if $\mathfrak{a}\subset\cup_{i=1}^{s}\mathfrak{p_i}$, then $\mathfrak{a_1}\subset \mathfrak{p_i}$ for some $i$

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg Background Definition 63 (Prime Ideal).: An ideal $\mathfrak{p}\neq R$ in a ring $R$ is a prime ideal if $rs\in ...
Seth's user avatar
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0 votes
2 answers
33 views

Relationship Between Subgroups of Abelian Groups & Ideals/Rings.

Clarification. I am currently reading from Dummit and Foote. Given $R$-module $M,$ we require $(1)$ $R$ is unital, and $(2)$ $1\cdot x=x$ for all $x\in M.$ When discussing rings $R,$ for the purposes ...
JAG131's user avatar
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Does the constant term of the minimal polynomial of an algebraic function also have no univariate factor if the algebraic function has none?

Each algebraic function is defined by an irreducible algebraic equation or its minimal polynomial, respectively. My question is: Let ${}^{-}$ denote the algebraic closure, $\pmb{n\in\mathbb{N}_{>1}}...
IV_'s user avatar
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0 votes
0 answers
36 views

Is $B$ flat as an $A$-module?

Suppose $A$ is an integral closed domain, and its quotient field is $K$. Suppose $L$ is a finite separable extension of $K$, and $B$ is the integral closure of $A$ in $L$. Is $B$ flat as an $A$-...
Born to be proud's user avatar
1 vote
0 answers
37 views

Are determinantal ideals Cohen-Macaulay?

The ideal generated by all the $t$-minors of an $m\times n$ matrix $X$, like $$ X=\begin{pmatrix} X_{11} & X_{12} & \dots & X_{1n}\\ X_{21} & \ X_{22} & \dots & X_{2n}\\ \...
Hola's user avatar
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0 answers
17 views

When a quotient of $\mathbb Z_p[[x,y]]$ is a regular local ring

Consider the power series in two variables with p-adic coefficients. Namely consider $\mathbb Z_p[[x,y]]$. Moreover let $c\in\mathbb Z_p$ and construct the quotient: $$ W=\mathbb Z_p[[x,y]]/(xy-c) $$ ...
manifold's user avatar
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0 votes
1 answer
58 views

Why $P(y/x) = 0$?

In Stack Project Commutative Algebra 119.2, $(R,m)$ is a local Noetherian ring. In one case of the proof, $x \in m$ is a nonzerodivisor, $y \in R$ and $ym \subset xm$. Then we have a map $$ \varphi: m ...
Functor's user avatar
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0 answers
23 views

A problem of determining whether a power series belongs to $\mathbb{C}(u)$

I am reading a paper "Drinfeld coproduct, quantum fusion tensor category and applications" and I have a probelm. Here is the arxiv:Drinfeld coproduct, quantum fusion tensor category and ...
fusheng's user avatar
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0 answers
36 views

Question about finding generators of the kernel for a substitution maps in rings.

The following is taken from Abstract Algebra A Comprehensive Introduction by: Lawerence and Zorzitto. Background Exercsie 6: Find generators for the kernel of each of the following substitution maps: ...
Seth's user avatar
  • 3,565
3 votes
0 answers
72 views

Estimation of the eigenvalue of a matrix

Let $A_n$ be a $n×n$ matrix with entries $a_{ij}=n-|i-j|$, let $\lambda_n$ be the largest eigenvalue of $A_n$, find $\lim_{n \to \infty} \frac{\lambda_n }{n^2}$ I find it hard to compute the ...
ddk's user avatar
  • 31
0 votes
0 answers
32 views

Proof that the order of elements stays the same under Frobenius endomorphism

Let $E: Y^2=X^3+Ax+B$ be an elliptic curve, defined over $\mathbb{F}_p$ where $p$ is a prime. Define: $$\phi: E(\bar{\mathbb{F}}) \rightarrow E(\bar{\mathbb{F}})$$ by $$\phi(P) = \left\{ \begin{array}...
Yvonne's user avatar
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1 vote
0 answers
26 views

Projective modules over group rings and trace

Consider a finite extension of complete valued fields $L/K$ with corresponding extension of rings $B/A$. Suppose $L/K$ is Galois with group $G$. I want to show that $B$ is a projective $A[G]$ module ...
Little Narwhal's user avatar
-2 votes
0 answers
81 views

Questions about a proof of a theorem on prime ideals.

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg Background Definition 63 (Prime Ideal).: An ideal $\mathfrak{p}\neq R$ in a ring $R$ is a prime ideal if $rs\in ...
Seth's user avatar
  • 3,565
1 vote
1 answer
34 views

Example of a residuated prelinear lattice that isn't linear

A residuated lattice is an algebra $$(L,\land, \lor, \star,\Rightarrow,0,1)$$ with four binary operations and two constants such that $(L,\land,\lor,0,1)$ is a lattice with the largest element 1 and ...
MtSet's user avatar
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0 votes
0 answers
34 views

Generic Dihedral group identity for rotations and reflections [closed]

From the wikipedia page I was able to see that the group structure has relationships between rotations and reflections $$ r_i s_j = s_{i+j} $$ However, it is not exactly clear to me how to show this ...
akozi's user avatar
  • 119
2 votes
1 answer
54 views

Signed measure and bounded total variation on algebras

Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable basis of the topology of $X$, which we can assume to be closed under finite unions and intersections, ...
Ennio's user avatar
  • 21
1 vote
0 answers
40 views

Conceptual definition of the Auslander-Reiten translate

In homological algebra, we learn to differentiate between The conceptual definition. A computation, which is done by choosing efficient resolutions. The only definition I've seen of the Auslander-...
user135743's user avatar
0 votes
0 answers
54 views

How to show normality is preserved under etale morphism?

Let $f:X\to Y$ be an etale morphism of Noetherian schemes, and $Y$ a normal scheme. How to show $X$ is a normal scheme? For $x\in X$, I only know that $\mathcal O_{Y,f(x)}\to \mathcal O_{X,x}$ is ...
Born to be proud's user avatar
1 vote
1 answer
26 views

Image of submodule to a quotient

Let $M,M'$ be $A$ modules. Let $N\le M$ be a submodule also $M' \le M$ is a submodule. Let $f:M\to \frac{M}{M'}$ be the natural $A$ module homomorphism. Claim, $f(N)= {(N+M')}/{M'}$ I wish to know if ...
Dinesh's user avatar
  • 786
0 votes
1 answer
41 views

Can restriction map trivialize a element in $H^1(G_K,M)$ by finite extension?

Let $K$ be a field and $G_K=\text{Gal}(\overline{K}/K)$ be absolute Galois group of $K$. Let $M$ be a finite $G_K$ module. For finite Galois extension $L/K$, let $H^1(G_K,M)\to H^1(G_L,M)$ be ...
Poitou-Tate's user avatar
  • 6,266
1 vote
1 answer
83 views

Automorphism of order 2 of $D_q$

Let $q\geq 3$ be a prime number and $D_q$ be the Dihedral Group of order $2q$. Find all automorphism of $D_q$ of order $2$. I tried this using a 'generic' automorphism $\varphi$ such that $\varphi(r)=...
Thomas García Villar's user avatar
4 votes
1 answer
52 views

Confusion about rings and ideals and how ideals are always subrings.

I've been studying rings and ideals recently and was told that all ideals are subrings. However, I seem to have been able to do the following to show that leads to an ideal being the ring itself. That ...
Timothy Bennett's user avatar
2 votes
1 answer
80 views

Group homomorphism between multiplicative groups of fields

Let $\mathbb{K}$ and $\mathbb{F}$ be two (algebraically closed) fields. I don't know what a group homomorphism $\mathbb{K}^*\to \mathbb{F}^*$ would look like. It is easy to see that $x\mapsto x^n$ ...
Eric's user avatar
  • 517
0 votes
1 answer
34 views

Method to find algebraic elements

Which of following is true for x$\epsilon$ R $ x^3$ is algebraic over Q implies x is algebraic over Q There is an x such that x is algebraic over Q($\sqrt 2$) but need not be algebraic over Q There ...
Alp1091's user avatar
  • 19
0 votes
1 answer
107 views

number of homomorphism from $K_4\to S_4$ [duplicate]

I want to calculate number of homomorphism from $K_4\to S_4$. let $\phi:K_4\to S_4$ be homomorphism then for each element $x\in S_4$ WHERE $|x|=2$ we have 3 cases , $ord(\phi(a))=1$ and $ord(\phi(b))...
Ricci Ten's user avatar
  • 136
1 vote
0 answers
69 views

For $R=\mathbb{k}[x,y]$, $\operatorname{Hom}_R(M,R)$ is always free.

Let $R=\mathbb{k}[x,y]$ be the polynomial ring with two variables over a field $\mathbb{k}$. I claim that $\operatorname{Hom}_R(M,R)$ is always free for any finitely generated $R$-module $M$. I ...
Ramanasa's user avatar
  • 450
0 votes
0 answers
28 views

Irreducible elements, prime elements, prime ideals, and maximal ideals [duplicate]

I'm trying to get the four concepts listed in the title straight in my mind and elucidate the relationship between them. Could someone check the following statements and let me know if they are all ...
Damalone's user avatar
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