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Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.

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Why is the following exercise is an improvement on a Lemma?

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg Background Lemma 65: Let $\mathfrak{a}$ be an ideal and let $\mathfrak{p_i}$ be prime ideals for $i=1,\ldots,s$. ...
Seth's user avatar
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Question about a proof for: ...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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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
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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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Generic Dihedral group identity for rotations and reflections

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
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2 votes
1 answer
29 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
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1 vote
0 answers
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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
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31 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
19 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
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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
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2 answers
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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
-2 votes
1 answer
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Is $0^0=1$ in algebra? [duplicate]

I've recently heard a conversation between an algebra professor and a Ph.D in mathematics. The professor said that it "almost annoyed him" how people keep saying that $0^0$ is undefined, ...
Elvis's user avatar
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1 answer
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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
77 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
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1 answer
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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
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1 answer
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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
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0 answers
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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
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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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3 votes
1 answer
108 views

Invariant Properties of Isomorphic Rings

I'm a second year maths student, looking at Rings and Modules questions for my exam. A property $P$ of rings is invariant under isomorphism if whenever $R$ is a ring with property $P$, and $S$ is a ...
CatsAndDogs's user avatar
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1 answer
59 views

$\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$?

I am exploring whether the following assertion holds true in integral domains: Consider two elements $a$ and $b$ in an integral domain. Define the LCM of $a$ and $b$ and also define the GCD of all ...
Martin Geller's user avatar
1 vote
1 answer
139 views

Relation between direct sum $\bigoplus$ and restricted product $\prod'$ of Galois cohomology

This is a question about the relation between directed sum $\bigoplus_{v \in M_K} H^1(\text{Gal}(\overline{K}/K), M)$ and restricted direct product $\prod'_{v \in M_K} H^1(\text{Gal}(\overline{K}/K), ...
Poitou-Tate's user avatar
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3 votes
3 answers
306 views

Does this condition imply normality for a subgroup?

(Note: This question emerged out of mere curiosity in the definition of a normal subgroup.) Let $(G, \cdot )$ be a group, and let $H$ be a subgroup. Now, for each coset in the set of (left ) cosets, $...
Vivaan Daga's user avatar
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1 vote
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A criterion for proving that an algebraic extension $E \subset F$ of fields is normal.

Let $K \subset F$ be fields such that $F$ is an algebraic extension of $K$, if for all elements $\alpha \in F$ there is a field $K \subset E \subset F$ such that $\alpha \in E$ and $E$ is normal over $...
Donlans Donlans's user avatar
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Gorenstein k-algebra

What is relation between gorenstein k-algebra and a system of parameters of a Cohen-Macaulay k-algebra Is that true? a gorenstein k-algebra is the quotient of a Cohen-Macaulay k-algebra over a system ...
ehsan hadian's user avatar
5 votes
2 answers
210 views

Find a Group Given 2 Rows of a Cayley Table

I am working through Maxfield and Maxfield's Abstract Algebra and Solution by Radicals, and the following exercise is included after discussing permutations and Cayley's Theorem: Two rows of a certain ...
RHyp's user avatar
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gcrd and Associates of an element of the Quaternion algebra over a totally real number field $K$

Let $K$ be a totally real number field of class number 1, and $Q$ the quaternion algebra over the ring of integers of $K$ with basis $\{1,i,j,k\}$ such that $i^2 = j^2 = k^2 = -1$ and $ij = -ji, ik = ...
Don Freecs's user avatar
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58 views

Ideal $\langle 3,x-1,y-2\rangle$ in $\mathbb{Z}[x,y]$

I am studying a little bit of ideals and come up with the exercise to show that the ideal $\langle 3,x-1,y-2\rangle$ is not equal to $\langle 1\rangle$ in the polynomial ring $\mathbb{Z}[x,y]$. At ...
Yeipi's user avatar
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0 votes
1 answer
36 views

Null vector does not depend on operations

Let S be a set and K a field. Is the following statement valid? If $V_1$ is a vector space constructed over K using S as its set of vectors (i.e. $(S,+ _1,._1)$ ) and $V_2$ is another vector space ...
LENIN YASSEL TRINIDAD FLORES's user avatar
-4 votes
0 answers
45 views

Sylow 3 subgroup in UT(2, Z7) [closed]

am studying group G= UT(2,Z_7) which has order n = 2^23^27 so there must be a Sylow 3-subgroup of order 9. But i can't figure out what this subgroup (call it H) must look like.
Gregor's user avatar
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4 votes
3 answers
412 views

Rank of a matrix over integers

What is the rank of a matrix whose entries come from the set of integers? Is it still the number of independent columns? For instake, take the matrix $$\begin{bmatrix} 6 & 3 & 2 \\ 12 & 6 &...
JD Maximo's user avatar
1 vote
1 answer
29 views

Number of Intersection Points of two Plane Polynomial Curves over $\Bbb R$.

For two polynomials $f,g\in\Bbb R[x,y]$ is it always true that the number of intersection points of the graphs of $f$ and $g$ is at most $\deg(f)\deg(g)$? Writing this out in standard notation, since $...
user108580's user avatar
2 votes
1 answer
29 views

The diagonalizable operators are dense in a finite complex vector space [duplicate]

I've been struggling with this problem from Axler's Linear Algebra Done Right (this is problem 11 from section 7F) and was looking for some hints. Problem: Suppose $\mathbb{F}=\mathbb{C}$ and $T \in \...
brianjsl's user avatar
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0 answers
25 views

Reference for the subgroup structure of PGL2(q)

I have already read the material for the classification of PSL2(q), but I cannot find the full classification of PGL2(q), hence I cannot justify whether I am right or not. If anyone knows where I can ...
Shen Jiahui's user avatar
0 votes
1 answer
57 views

Number of group actions

Suppose that we have an action of a group $G$ on a $\Bbb{Z}$. This action corresponds to a homomorphism $\varphi: G \to Aut(\Bbb{Z}) \simeq C_2$. But we can have many homomorphisms of $G$ on $C_2$. Is ...
Greg's user avatar
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5 votes
0 answers
54 views

The trace of a matrix and the radical ideal generated by its power

Consider the polynomial ring $ \mathbb{C}[x_{11}, \dots , x_{nn}] $ with $ n^2 $ variables and the matrix $$ A := \begin{pmatrix} x_{11} & \cdots & x_{1n} \\ \vdots & & \vdots \\...
Long-Ping Li's user avatar
0 votes
0 answers
36 views

automorphism group isomorphic to $\mathbb{Z}/2\mathbb{Z}$ [duplicate]

When is the automorphism group of a group $G$ isomorphic to $\mathbb{Z}/2\mathbb{Z}$ ? I noticed that the only groups whose automorphism group is trivial are trivial one and $\mathbb{Z}/2\mathbb{Z},$ ...
praton's user avatar
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2 votes
1 answer
32 views

Is restriction map $\text{Hom}(S_3, \Bbb{Z}/2\Bbb{Z}\times \Bbb{Z}/2\Bbb{Z})\to \text{Hom}(H_2,\Bbb{Z}/2\Bbb{Z}\times \Bbb{Z}/2\Bbb{Z})$ injective?

Let $S_3$ be symmetric group of order $6$. Let $H_2$ be 2 Sylow subgroup $H_2=\{1,(12)\}$. Is restriction map $\text{Hom}(S_3, \Bbb{Z}/2\Bbb{Z}\times \Bbb{Z}/2\Bbb{Z})\to \text{Hom}(H_2,\Bbb{Z}/2\...
Poitou-Tate's user avatar
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1 vote
1 answer
36 views

Is restriction to $p$ Sylow subgroup $\text{res} : H^1(H,M)\to H^1(H_p,M)$ injective?

'Galois cohomology of Algebraic number fields' written by K. Haberland reads the following lemma in page 66. Let $H$ be a finite group. Let $p$ be a prime number and $H_p$ be a fixed p Sylow subgroup ...
Poitou-Tate's user avatar
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5 votes
2 answers
247 views

Is every group the semidirect product of its center and inner automorphism group?

For every group $G$, we have $$G/Z(G)\simeq \operatorname{Inn}(G).$$ I wonder whether the quotient projection has a right inverse. I suspect it doesn’t have one in general. But I can’t find a counter ...
user760's user avatar
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0 votes
0 answers
49 views

$\mathbb{Z}[x,y]$ Vs $\mathbb{C}[x,y]$ [duplicate]

Is the following statment true/false ? $I=(x+y,x-y)$ is a prime ideal of both $\mathbb{Z}[x,y]$ and $\mathbb{C}[x,y]$ My attempt :I think this statement is true $$\frac{\mathbb{Z}[x,y]}{(x+y,x-...
jasmine's user avatar
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0 votes
1 answer
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Is this hypercomplex system sound? If so what is it called?

sorry for the bad image quality, I had to improvise Definiton of a hypercomplex number [it's in the tag] : A hypercomplex number is an element of a finite-dimensional algebra over the real numbers ...
AnonymousPoster's user avatar
2 votes
0 answers
66 views

$\overline{\mathbb{F}_2}$ does not contain a primitive 10th root of unity

I need to prove/disprove the following statement: Every algebraically closed field $K$ contains a 10th root of unity. I don't think the statement is true. My counterexample is as follows: Let's take ...
muhammed gunes's user avatar
0 votes
1 answer
40 views

Justification of wedge-determinant equality

In Do Carmo's Curves and Surfaces, he states the following: Can someone explain the reasoning here? And what is meant by for all $i,j,k,l = 1,2,3$? Aren't the $e$'s meant to be the three different ...
DC2974's user avatar
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4 votes
1 answer
50 views

Rotations in $C\ell_{16,0}$

Consider the Clifford algebra $C\ell_{8,0}$. If $x\in C\ell_{8,0}$ and $u$ is a spinor in $C\ell_{8,0}$, then the rotations in this algebra can be written as $$ x' = u x u^{-1}. $$ Now consider $C\...
user38680's user avatar
7 votes
1 answer
116 views

A representation ring of all finite groups

Let $G_1$, $G_2$ be finite groups. Let $V_1$ and $V_2$ be finite-dimensional complex representations of $G_1$ and $G_2$, respectively. Then their tensor product naturally becomes a representation ...
Smiley1000's user avatar
  • 1,509
1 vote
0 answers
20 views

on the definition of prime ideals [duplicate]

An ideal $P$ of $R$ is said to be a prime ideal if and only if $P≠R$, and whenever $A$ and $B$ are ideals of $R$, then $AB⊆P$ implies $A⊆P$ or $B⊆P$. but what if $A⊆P$ and $B⊆P$? is it still a prime ...
Isaac 's user avatar
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2 votes
0 answers
18 views

Jordan decomposition of $GL_n$ over perfect fields

Matrices in $g\in GL_n$ over a perfect field $k$ have a (unique) Jordan decomposition, i.e. there exist unique $g_u$ and $g_s$ such that $g=g_u\cdot g_s = g_s\cdot g_u$ where $g_u$ is unipotent ($(g_u-...
Vincent Batens's user avatar
-1 votes
1 answer
37 views

Example of non-unital ring where the additive neutral element is not asorbing for multiplication

I can prove that in every unital ring (or more generally in every unital semiring where the additive monoid is cancellative) the additive neutral element 0 is absorbing, meaning that a.0=0=0.a for ...
Gérard Lang's user avatar
5 votes
1 answer
138 views

Reflexive objects in the category of $G$-sets

Let $G$ be a group. Consider the category of $G$-sets, whose objects are sets $X$ with an action $G \times X \to X$ satisfying $e \cdot x = x$ and $g \cdot (h \cdot x) = (gh) \cdot x$ and whose arrows ...
Mockingbird's user avatar
  • 1,037
2 votes
1 answer
68 views

If $G$ is a group with each nontrivial element having an infinite order, does $\mathbb{Z}[\mathbb{G}] \cong 0$?

The definition of group ring in Wikipedia is Let 𝐺 be a group, written multiplicatively, and let 𝑅 be a ring. The group ring of 𝐺 over 𝑅, which we will denote by 𝑅[𝐺], or simply 𝑅𝐺 is the set ...
Jiahao Li's user avatar

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