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.
85,114
questions
8
votes
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Example of a cocommutative, non-unimodular Hopf algebra?
1. Definitions: Unimodularity and cocommutativity
Let $H$ be a Hopf algebra over a field $\mathbb k$.
We call $H$ unimodular if the space of left integrals $I_l(H)$ is equal to the space of right ...
- 3,018
2
votes
1
answer
134
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Two (different ?) definitions of a Gröbner-Basis
I have two slightly different definitions for Gröbner-Bases.
1.Definition from book
Let $I$ be an ideal and $G=(g_1,\ldots,g_s)$ a basis for $I$. $G$ is called a Gröbner-Basis if $\langle LT(g_1),\...
- 351
4
votes
1
answer
949
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Prove quotient of graded ring with graded ideal is a graded ring
Let $S=\oplus S_i$ be some graded ring and let $I\subset S$ be a graded/homogeneous ideal of $S$. That is to say, $I=\oplus I_i$, where $I_i=S_i\cap I$ (this is equivalent to the property that $I$ has ...
- 3,036
17
votes
1
answer
811
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Integrals of a Hopf algebra: Why that name?
1. Context: The notion of an integral
Let $H$ be a Hopf algebra over a field $\mathbb k$. We call its $\mathbb k$-linear subspace
$$
I_l(H)= \{x \in H; h \cdot x=\epsilon(h)x \quad for \>all\>h\...
- 3,018
1
vote
1
answer
57
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Is $(p)$, $p$ prime, ideal prime in $\mathbb{Z}[x,y]$?
Is correct the following? I want verify if $(p)$ is prime in $\mathbb{Z}[x,y]$ ($p$ prime) I have this: $\mathbb{Z}[x,y]/(p)\simeq (\mathbb{Z}[x]/(p))[y]\simeq (\mathbb{Z}_{p}[x])[y]$ and $\mathbb{...
- 3,548
1
vote
1
answer
174
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Compute the order of each of the elements in $D_6$ where $D_{6}=\left\langle r, s \mid r^{3}=s^{2}=1, r s=s r^{-1}\right\rangle$
Compute the order of each of the elements in $D_6$ where $D_{6}=\left\langle r, s \mid r^{3}=s^{2}=1, r s=s r^{-1}\right\rangle$
I found six elements of $D_6$ are $1,r, r^2,s, rs, r^2s.$
How can I ...
- 163
-1
votes
2
answers
199
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What does a two by two union of disjoint sets mean in $A_1, A_2, A_3$, for every pair $i,j \in {1, 2, 3}$ and $i \neq j$? [closed]
I know the definition of a disjoint set, but it is the first time that I have heard about a two by two union in three sets $A_1, A_2, A_3$ when
$\cap_i^3 A_i$ $=\varnothing$ and $A_i \cup Aj \neq \...
- 230
5
votes
1
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152
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Showing whether an ideal in $\mathbb{Z}[x,y]$ is prime.
The ideal $(1+x^2,1+y^2)$ is prime in $\mathbb{Z}[x,y]$?
I have this:
Analogously to $\mathbb{Z}[x]/(1+x^2)\simeq \mathbb{Z}[i]$, $\mathbb{Z}[x,y]/(1+x^2,1+y^2)\simeq \mathbb{Z}[i]\times \mathbb{Z}[i]$...
- 3,548
-1
votes
1
answer
154
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Cardinality of double cosets
We show that the sets of double cosets $K\backslash G/H$ is in bijection with $K\backslash(G/H)$. So $\vert K\backslash(G/H) \vert = \vert G/H \vert /\vert K \vert = (\vert G \vert / \vert H \vert)/\...
- 4,206
1
vote
2
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1k
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Remainder Theorem in complex numbers.
Is the Remainder Theorem works in complex numbers? I have read many articles but they just say linear function , not include whether complex numbers are allowed. Can someone check my understanding ...
- 1,223
-1
votes
2
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582
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Prime Ideal with 1
I know that it is possible for a prime ideal $P$ to not contain $1$ (the even numbers are a prime ideal of $\mathbb{Z}$), but I can't figure out if every prime ideal does not contain $1$, and I can'...
- 1
1
vote
0
answers
47
views
What are some less known pairs of binary operators on Integers are distributive?
The operations of addition(+) and multiplication(*) as ordinarily defined on integers are distributive.
$$a*(b+c) = a*b + a*c $$
There are a lot of other sets on which operations defined on them are ...
- 31
2
votes
0
answers
200
views
`Normalization of $\mathbb{P}^1_k $ in the fraction field of a hyperbola
(This is Ravi Vakil Exercise 9.7.L(b))
I’m trying to describe the normalization of $$\mathbb{P}^1_k = \operatorname{Spec}(k[x]) \cup \operatorname{Spec}(k[x^{-1}])$$ in the fraction field, $F$, of $$k[...
0
votes
1
answer
104
views
Why is $6$ the multiplicative identity of the ring $2 \Bbb Z_{10}$?
Just wondering how you are able to determine that the multiplicative identity of the ring $2 \Bbb Z_{10}$ which is $\{ 0,2,4,6,8 \}$ is 6. I tried multiplying every element in this ring by 6, but I ...
- 161
1
vote
1
answer
467
views
Clarification in a proof that free modules are flat
In section 10.5 of Dummit and Foote, we are given a proof that free modules are flat that goes like this:
1). Finitely generated free modules are flat (easy)
2). Suppose now that $F$ is an arbitrary ...
1
vote
1
answer
312
views
Group of units of $C[0,1]$
Is the group of units of $C[0,1]$ cyclic?
I think it is not cyclic. The first argument that came to my mind is that if it is cyclic then its generator must be a constant function, but not all units of ...
- 979
1
vote
1
answer
230
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Ordered Groups: Left Multiplication vs Right Multiplication
Given that $G$ is a linearly ordered group (bi-ordered). I want to try and understand the difference between the “size” of left multiplication vs right multiplication (which I have written below using ...
- 1,304
1
vote
2
answers
141
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Meaning of an element "vanishing" on $V(I)$
I am reading a question on Vakil's algebraic geometry notes and I am confused by a particular term (exercise 3.4.J in the November 17, 2018 draft on page 118).
Suppose $I\subset B$ is an ideal where $...
- 2,666
2
votes
1
answer
100
views
Inconsistency in standard definition of abstract vector spaces?
A vector space $V$ over field $F$ is a nonempty set with two binary operations, "addition" and "multiplication." Addition is a function $V \times V \rightarrow: (v_1, v_2) \...
- 948
1
vote
0
answers
89
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𝑆𝑝𝑖𝑛(8) 8 dimensional representations and mutation
Here is a follow up of on strange facts of $Spin(8)$ group Decompose $SO(8)$ and $Spin(8)$ 8 dimensional representations to $SO(m)$ and $SO(n)$
We know that the $Spin(8)$'s vector representation is 8 ...
- 3,667
1
vote
0
answers
360
views
Proving $\mathbb{Z}_p[x]$ has Unique Factorization.
I'm self-learning some number theory. I'm trying to prove or disprove the following:
$$\mathbb{Z}_p[x]\text{ has a Unique Factorization Theorem}$$ (where $p$ is some rational prime)
Consider some $f \...
- 331
0
votes
1
answer
49
views
Prove or disprove that the given group is abelian
Let $K=K_1 \cup K_2 \cup \dots \cup K_n$ be a finite union of tori in 3-space (each $K_i$, $i=1,2,\dots, n$ is a torus). For $i=1,2,\dots, n$, let $a_i$ be the meridian and let $b_i$ be the longitude ...
- 1,101
4
votes
1
answer
249
views
Example where $\operatorname{Spec} S^{-1}B$ is neither open nor closed in $\operatorname{Spec} B$
I know that $\operatorname{Spec} S^{-1}B$ is open in $\operatorname{Spec} B$ with respect to the Zariski topology when $S=\{1,f,f^2,\ldots\}$ for $f\in B$.
However, is this true for every ...
- 2,666
2
votes
2
answers
133
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Equivalent properties of graded ideals
Let $S=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$. Note that $S=\oplus_{i\in\mathbb{N}}S_i$ where $S_i$ is the $k$-space spanned by all monomials of total degree $i$. Therefore any $f\in ...
- 3,036
2
votes
1
answer
104
views
Simple reduced rings which are not domains
a ring $R$ is reduced if for each $a\in R$ , $a^n=0$ implies that $a=0$ for any positive integer $n$. also $R$ is called simple if it doesn't have any proper two-sided ideal.
Is there any example of a ...
user752594
1
vote
3
answers
252
views
Multiplication group for $\mathbb Z_n$ modulo $n$
By definition:
Let $\mathbb Z^+_n = \{[0],[1],[2],\ldots,[n−1]\}$
$\mathbb Z^+_4 = \{[0],[1],[2],[3]\},$
but
how $\mathbb Z^*_{12}$ is $\{[1],[5],[7],[11]\}$ ?
how $\mathbb Z^*_{7}$ is $\{[1],[2],[3],[...
- 151
1
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1
answer
209
views
Do algebraic fields in mathematics have any connection to the fields in physics? [duplicate]
Basic question, but is there some sort of connection between the two, or are they just separate definitions.
1
vote
1
answer
86
views
Taft-Hopf Algebra has dimension $N^2$?
Definition of the Taft-Hopf Algebra
Let $k$ be a field. Let be $N$ a positive integer such that there exists a primitive $N$-th root of unitiy $\zeta$ over $k$.
Denote by $(H, \mu, 1_H)$ the unital, ...
- 3,018
0
votes
1
answer
275
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Why isn't Universal enveloping algebra graded?
Given a Lie algebra $L$, define $U(L) = T(L)$ mod $I(L)$ where $T(L)$ is the tensor algebra of $L$ and $I(L)$ is the two sided ideal of $T(L)$ generated by all elements of the form $xy-yx-[x,y]$ where ...
user637978
2
votes
2
answers
56
views
Maximal Ideals in a UFD
Consider the Ideal $I=(ux,uy,vx,uv)$ in the polynomial Ring $\mathbb Q[u,v,x,y]$, where $u,v,x,y$ are indeterminates. Prove or disprove that every maximal Ideal $M$, containing $I$ contains the Ideal $...
- 2,136
3
votes
1
answer
298
views
$A,B$ Noetherian rings, $A\subseteq B$ integral extension, $\mathfrak m $ a maximal ideal of $A \implies B/\mathfrak m B$ is Artinian
Let $A,B$ be Noetherian rings, $A \subseteq B$, such that $B$ is integral over $A$. Given $\mathfrak m\subseteq A$ a maximal ideal, prove that $B/\mathfrak mB$ is an Artinian ring.
I'm really stuck.
...
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0
votes
0
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16
views
Does the component of a vector in $L_{2}$ need to be within whatever limits I place upon it?
In $L^{2} (a,b)$ let there be a vector $V(x_{1},x_{2},\dots)$. We have $x_{i}$ defined as:
$$x_{i}\ = \ \langle e_{i}|V \rangle$$
Where $e_{i}$ is the orthonormal basis set. Since $V$ is a part of $L^...
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1
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1
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116
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Is $\operatorname{Aut}(D_{12})\simeq D_{12}$?
Let $D_{12}$ be the dihedral group of order 12. Then
$$|\operatorname{Aut}(D_{12})|=6\phi(6)=12=|D_{12}|,$$
and the standard method of proof for
$$\operatorname{Aut}(D_6)\simeq D_{6}\qquad\mbox{and}\...
- 6,946
3
votes
2
answers
142
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Why is $[\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}] \le 4$
I am working on the following exercise:
Show that $\mathbb{Q}(\sqrt{2},\sqrt{3})$ has degree $4$ over $\mathbb{Q}$ by showing that $1,\sqrt{2},\sqrt{3}$ and $\sqrt{6}$ are linearly independent.
I ...
- 4,178
3
votes
1
answer
302
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Maximal and prime ideal of $R:=\prod\limits_{n=1}^\infty \mathbb{Z}/m\mathbb{Z}$
Let $m$ be an integer such that $m \ge 2$. We define $R$ as the countable direct product
of the ring $\mathbb{Z}/m\mathbb{Z}$
$$R:=\prod_{n=1}^\infty \mathbb{Z}/m\mathbb{Z}$$
I am trying to prove that ...
- 31
5
votes
1
answer
172
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The set of elements of order $7$ of $A_7$ is not a conjugacy class.
I think the problem is simple but I just want to make sure I am doing it right.
Elements of order $7$ are of the form $(abcdefg)$. There are $6!$ such elements. But $6! \nmid 7!/2$ and thus if it was ...
- 3,286
0
votes
1
answer
100
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Formal way of proving that the ring $2 \Bbb Z_{10}$ has an additive inverse
I'm trying to find a formal way of proving that the ring $2 \Bbb Z_{10}$ has an additive inverse. I understand that 8+2 = 0 mod 10 and 6+4 = 0 mod 10, etc. Is there a way to formally prove this is ...
- 161
2
votes
1
answer
75
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Find the order of $\sqrt[3]{2}(1+\zeta_3)$ and $\sqrt[3]{2}(1-\zeta_3)$ over $\mathbb{Q}$
Find the order of $\sqrt[3]{2}(1+\zeta_3)$ and $\sqrt[3]{2}(1-\zeta_3)$ over $\mathbb{Q}$
I am not sure how to tackle this problem.
MY notes:
$\mathbb{Q}(\sqrt[3]{2},\zeta_3)$ has degree 6. The degree ...
- 3,286
1
vote
1
answer
367
views
Existence of a ring homomorphism implies characteristic divisibility
Let $A$ and $B$ be rings having respective characteristics $n$ and $k.$ Show that if there exists a ring homomorphism $f : A \to B$, then $k$ divides $n$.
I don't know how to start this problem. If ...
user775659
4
votes
1
answer
209
views
Irreducibility over finite fields
I am trying to do the problem: is $\mathbb{F}_{2011^2}[x] /(x^4 -6x -12)$ a field?
I know that this is a field if and only if $(x^4 - 6x - 12)$ is a maximal ideal, if and only if $x^4 - 6x - 12$ is ...
- 91
2
votes
1
answer
70
views
Show that $ker(f) = n/k$
We suppose that we have integers, $n$ and $k$, such that there exists a ring homomorphism defined as : $$f : \mathbb{Z}/n\mathbb{Z} \rightarrow \mathbb{Z}/k\mathbb{Z}$$ where $\overline{x} \...
user775659
3
votes
3
answers
78
views
Homomorphism from $\mathbb{Z}_{120}$ to $\mathbb{Z}_{167}$
Find all the morphisms from $\mathbb{Z}_{120}$ to $\mathbb{Z}_{167}$.
I've tried that:
Let $f : \mathbb{Z}_{120} \to \mathbb{Z}_{167}$ a homomorphism.
Then let $x \in \mathbb{Z}_{120}$, then $x^{120} =...
- 446
3
votes
1
answer
71
views
Antipode of a Hopf algebra satisfies $\sum_{(x)} S(x') \otimes x'' \otimes x''' = \sum_{(x)} \epsilon(x') \otimes x''$
Page 51 of Christian Kassel’s “Quantum Groups” contains the following:
Let $(H, \mu, \nu, \Delta, \epsilon, S)$ be a Hopf Algebra. Then for all $x$ in $H$ we have $\sum_{(x)} S(x') x'' = \epsilon(x) ...
user637978
4
votes
2
answers
577
views
Simple exercise on a linear operator $T$
I'm given the following linear operator $$T(a,b)=(-2a+3b,-10a+9b)$$ on the vector space $V=\mathbb{R}^2$. I have to find the eigenvalues of $T$ and an ordered basis $\beta$ for $V$ such that $[T]_\...
user812250
1
vote
3
answers
142
views
Modules which are both left and right modules over a noncommutative ring
Given a noncommutative ring $R$, is there a name for those (left) $R$-modules $M$ for which
$$
m.(rs) = m.(sr), ~~~ \textrm{ for all } r,s \in R, \textrm{ and } m \in M?
$$
(Note that we have denoted ...
- 187
1
vote
1
answer
114
views
Constructing $(\Bbb N,+)$ via Peano function algebra duality.
In the next section we outline a $\text{ZF}$ construction of the natural numbers under addition using a 'duality' argument (the choice of the word duality is subjective and has no formal meaning).
...
- 11.4k
0
votes
2
answers
94
views
Normalizers of Sylow $p$-subgroups have restricted order
I saw an old post. It gives the following result.
Assume that $H$ is a subgroup of a finite group $G$, and that $P$ is a Sylow $p$-group of $H$. If $N_G(P) \subset H$ then $P$ is a Sylow p-subgroup ...
user792898
5
votes
1
answer
114
views
Non-simplicity of Frobenius complements
I'm reading a paper and it says the following theorem implies that the Frobenius complement of any finite Frobenius group is not a non-abelian simple group.
(Zassenhaus 1936) Let $G$ be a finite ...
user792898
3
votes
1
answer
394
views
The advantage of Complex Differentiation and Inverse Function Theorem
One interesting phenonmenon in complex analysis is the following,
If $f:\mathbb C\to\mathbb C$ is complex differentiable at point $a$ ($\equiv$derivative is a spiral similarity), and a local ...
user376921
1
vote
1
answer
151
views
Does a group homomorphism always produce a group extension.
We all know the very famous group isomorphism theorems:
If $\varphi:G \rightarrow H$ is a group homomorphism them $G/\ker(\varphi)$ is isomorphic to $im(\varphi)$.
If $f: X \rightarrow Y$ is a ...
- 820