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.
87,894 questions
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The set $\mathcal{S}(R)$ of all radical ideals of a ring $R$ might form a boolean ring with $\circ=$ ideal addition and $\oplus=$ a certain quotient.
Define $S \subset \Bbb{N}$ to be the square-free integers $s$ i.e. such that no $p^2 \mid s$ for any $p \in \Bbb{P}$ a prime number.
Then it is easy to see that $(S, \oplus)$ forms an boolean group ...
- 22.7k
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Tricks for Computing the Center of a Group
I was doing a homework question about computing the center of a group, and realized everytime I've ever computed the center, I am very explicitly writing down elements and finding restrictions.
I ...
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Prove that $\mathbb{Q}(\alpha, \beta, \gamma, r_1) = \mathbb{Q}(r_1, r_2, r_3, r_4)$
Problem Statement: Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$ be an irreducible polynomial over the field of rational numbers $\mathbb{Q}$, where $a, b, c, d \in \mathbb{Q}$. Let $r_1, r_2, r_3, r_4$ be ...
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Irreducible elements in a direct product of rings $R_1 \times \cdots \times R_n$
Let $R_1, \dots, R_n$ be rings and consider their direct product $R = R_1 \times \cdots \times R_n$. What do the irreducible elements of $R$ look like?
To get some intuition, I started with the ...
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Defining a 2D TQFT that maps $S^1$ to the vector space of of all $n×n$ matrices over $\Bbbk$.
Searching for some simple examples of TQFTs I found the following among the Exercises of Kock's "Frobenius algebras and 2D Topological Quantum Field Theories".
There Kock considers a TQFT $ ...
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Proving k(α) is a finite extension. [closed]
Let k ⊆ k(α) be a simple extension, with α transcendental over k. Let E be a
subfield of k(α) properly containing k. Prove that k(α) is a finite extension of E.
This is a question from the book "...
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When is an open immersion affine as a morphism of schemes?
This might be a stupid question but I could not convince myself of the answer.
Let $j:X \to Y$ be an open immersion of schemes, and assume that $j$ is affine. Very broadly, what can we say? More ...
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Duality and projectively stable category
Let $A$ be a finite‑dimensional algebra over an algebraically closed field $k$. I work with right modules.
We have a duality given by $F=\operatorname{Hom}(_,A):\mathrm{mod}(A)\to\mathrm{mod}(A^{op})$....
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Morphisms in $\mathfrak{R}^3$ and $\mathfrak{R}^6$ with $\mathcal{mod}\,A$
I am working with representation theory of finite-dimensional algebras.
While studying powers of the radical of the module category, I found the following
definition (for indecomposable modules):
$$
\...
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Vector space of vectors [closed]
I have a question that is really confusing for me. In Serge Lang, the elements of $K^n$ where $K$ is a field are called vectors. The thing is that the vector $V=(v_1,...,v_n)$ belongs indeed to $K^n$ ...
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Additive Right Adjoint functor F preserves left exact sequence? Question about the proof
$f:A\to B$ and $g: B\to C$ are two morphisms of R-Mod. $0\to A \to B\to C\to 0$ is exact and F is an additive right adjoint functor.
I want to show $0\to F(A) \to F(B)\to F(C)\to 0$.
I know right ...
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Literature on Differential Forms of finite extensions of smooth $k$-algebras
Given a finitely generated smooth $k$-algebra $A$ (the case where $A=k[X_1,...,X_n]$, or a localization of this, is the one I am actually interested in) and a finite extension $B$ of $A$, I am ...
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Ncat's characterization of Real numbers [closed]
Hi! This is my first post so please be gentle ;).
I was reading this ncat page that reads as follows:
There is a characterisation of the real line as the ‘complete archimedean Tarski group’ due to ...
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Is a division ring with a compatible total order, necessarily commutative?
Let $D$ be a division ring with a total order $\le$ which is compatible with addition and multiplication, i.e. for any $a, b, c \in D$
\begin{align}
&a \le b &\implies& a + c \le b + c \\
&...
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If the evaluation map $\varepsilon\colon R[X] \to R^R$ is injective, does every non-zero polynomial have only finitely many roots?
Let $R$ be a (commutative, unital) ring. We then have a homomorphism of rings $\varepsilon \colon R[X] \to R^R$, given by associating to a polynomial the function it induces.
EDIT: More precisely, for ...
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Semivector spaces over commutative semifields
For the case of commutative field, we have that each vector space has a basis and even more, each linearly independent set can be completed into a basis. Do we have a same result for general ...
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Doubt on how much is unique a certain homomorphism into the modules of differential
Let $k$ be a ring and let $A$ be a $k$-algebra. Denote by $\Omega^1_{A/k}$ the module of derivations on $A$ over $k$.
Let $I\subset A$ be an ideal. The composition of $k$-linear homomorphisms $$I\...
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Does the tensor $\Phi$ take this form?
We will use Einstein summation convention. I apologize if this question is too easy—it has been years since I've had to to work with some of the mentioned material.
Have the the set of all linear ...
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Center of a finite perfect group
This question is distantly related to the following MathStack post: How "big" can the center of a finite perfect group be?
The above post and its answers comment on the size of the center of ...
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Name of the algebraic structure
I am looking for the name of an algebraic structure that generalizes the concept of a monoid. Suppose we have a set $ S $ with a binary operation $ + $ and binary operation $ * $ . They are both ...
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Description of the ideals of $A[X]$ using stationary chains of ideals of $A$
I am currently studying (commutative) ring theory, and recently finished proving the transfer lemma for noetherian rings (if $A$ is a noetherian ring, then its rings of polynomials $A[X]$ is also ...
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Classification of $C^{*}(G)$ for a compact group $G$
I want to prove that $C^{*}(G)\cong \oplus_{\pi\in\hat{G}}M_{dim\pi}(\mathbb{C})$. Of course, one wants to use the Peter-Weyl theorem for this (the version, which states that $L^{2}(G)\cong \oplus_{\...
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How applicable are the isomorphism theorems? [closed]
How broadly do the (four?) isomorphism theorems apply? Do they hold only of groups? What do they look like in set theory?
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If $K/F$ and $L/F$ are separable, then the composite $KL/F$ is separable
Prove that, if $K/F$ and $L/F$ are separable, then the composite $KL/F$ is separable.
My attempt: for finite extensions, this useful lemma holds:
Let $K/F$ be a field extension. If $\alpha_1, \dots, ...
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Examples of countable non-commutative division rings
There is a finite field for every power of a prime integer, the field of rational numbers is countable, and R and C are fields with continuum cardinals.
No non-commutative division ring can be ...
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Classification of Cyclotomic Skewfields?
I say that a skewfield (division ring) $D$ over $\mathbb{Q}$ is cyclotomic whenever $D$ admits a finite $\mathbb{Q}$-basis $\{\zeta_1,\dots,\zeta_n\}$ with each $\zeta_i$ a root of unity (i.e., $\...
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What can be said about $K((x))((y))\otimes_{K((x,y))} K((y))((x))$?
Let $K$ be a field. There is the iterated field of Laurent series
$$
K((x))((y))=\{f:\mathbb{Z}^2\to K:f(x,y)=0\,\text{for}\,y<-N\,\text{or}\,y\ge -N,x<-N_y\},
$$
and similarly $K((y)((x))$. ...
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Do division rings have algebraic closures?
The theorem of existence of the algebraic closure of a field can be written as the following assertion:
Let $Q$ be a field, then there exists a field $C$ (having the same characteristic than $Q$) ...
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The equivalent definitions of quasi-Frobenius ring
There are two equivalent definitions of the quasi-Frobenius ring:
(1)$R$ is Noetherian on one side and self-injective on one side.
(2)$R$ is Artinian on a side and self-injective on a side.
I know ...
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If $f(x)$ is irreducible over $\mathbb{Q}$, the equivalence classes of roots of $f(x)$ under transpositions of the Galois group are the same size
Let $f(x) \in \mathbb{Q}[x]$ be an irreducible polynomial over $\mathbb{Q}$ and let $R$ be set of roots of $f(x)$ in a splitting field $K$ (over $\mathbb{Q}$). Call $G$ the Galois group of $K/\mathbb{...
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Kernel zero but not monic?
Is there a category $\mathcal{C}$ with zero object and a morphism $f:A\to B$ such that $0:A\to A$ is a kernel but $f$ is not monic?
If $\mathcal{C}$ is preadditive then $\ker f = 0 \iff f$ is monic. ...
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Linear factors of polynomial in a polynomial ring
What is the fastest known method to split $f \in \mathbb{Z}[x]$ into linear factors mod $g \in \mathbb{Z}[y]$, assuming this happens?
Since that is not very clear, here is an example of what I'm ...
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Prove that the extension $\mathbb{Q} \subset \mathbb{Q}(\sqrt{3 + \sqrt{7}})$ is not normal
I want to prove that the extension $\mathbb{Q} \subset \mathbb{Q}(\sqrt{3 + \sqrt{7}}) = L$ is not normal. My strategy is to show that the minimal polynomial $f(x) = x^4 - 6x^2 + 2$ of $\alpha = \sqrt{...
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Show that $\{E:F\}\leq [E:F]$
I am doing the exercises of Fraleigh's book. Exercise 49, question 14.
Let $F$ be a field, $E$ be a finite extension of $F$. Let $\{E:F\}$ be the number of isomorphisms of $E$ to a subfield of $\bar F$...
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Chinese remainder theorem in rngs (rings without identity)
Hungerford states the Chinese remainder theorem as follows.
Let $A_1,\cdots,A_n$ be ideals in a ring R such that $$\begin{cases} R^2 + A_i = R\ \ \ \forall i \\ A_i+A_j = R\ \ \ \ \forall i \ne j \end{...
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Can I define automorphism on conjugate generators of a field?
Let $L/K$ be a field extension. Suppose $f(x)$ is the minimal polynomial of $\alpha$ and $\alpha'$ over $K$ and suppose $g(x)$ is the minimal polynomial of $\beta$ and $\beta'$ over $K$. If $L = (\...
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Main theorem of Galois Theory: professor Borcherds' counting argument
In this lecture, professor Borcherds gives a great proof of the main theorem of Galois Theory. My question is about the counting argument he uses to prove the following fact.
If $M/K$ is a Galois ...
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Which integral domains $R$ are flat over every subring?
It is well-known that if every integral domain containing a given integral domain $R$ is flat over $R$, then $R$ is a Prüfer domain. So I would like to ask the question about the other direction: ...
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What is an example of a finite, indecomposable semigroup with left identity and right inverses other than the basic example $g \ast h := h$?
This old question asks for an example of a semigroup $(G, \ast)$ for which
(a) $G$ has a left identity, i.e., an element $e \in G$ such that $e \ast g = g$ for all $g$, and
(b) every element of $G$ ...
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Confusion about one of Cartan-Weyl Basis rules for Lie algebra
(forenote: forgive my bad notation and my inability to move away from the word eigenvector - I'm still trying to wrestle with these concepts very crudely, so I don't want to use words I'm not ...
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Must the spectral map $\operatorname{Spec}B\to\operatorname{Spec}A$ be surjective if the image contains all closed points of $\operatorname{Spec}A$?
Let $A\to B$ be an injective homomorphism of commutative rings. If the image of the spectral map $f:\operatorname{Spec}(B)\to\operatorname{Spec}(A)$ contains all closed points of $\operatorname{Spec}(...
- 2,783
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Since rings of continuous maps exist why in (co)homology do you never see chain complex $C_n =$ certain ring of continuous functions "of degree $n$"?
I'm new to homological algebra. Just wondering why we never seem to see the involved chain complexes defined simply to be $C_n=$ continuous maps "of degree $n$" where the context makes the ...
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Are flat modules acyclic w.r.t. the completion functor?
Let $(R,\mathfrak{m})$ be a noetherian ring, and $\hat{R} = \varprojlim_i R/\mathfrak{m}^iR $ its $\mathfrak{m}$-adic completion. We may define the completion functor $\Lambda\colon \text{$R$-Mod} \to ...
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If $K/L$ is normal and $L/F$ is purely inseparable, then $K/F$ is normal - How to finish the proof?
This question is related to If $F/L$ is normal and $L/K$ is purely inseparable, then $F/K$ is normal .
Let $K/L$ and $L/F$ be field extensions. if $K/L$ is normal and $L/F$ is purely inseparable, ...
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How can we show this "Small Wall-Kervaire Braid Lemma"?
Consider the following diagram of abelian groups:
Assume that the $\color{blue}{\text{blue braid}}$ and the $\color{green}{\text{green braid}}$ are long exact sequences and that the $\color{red}{\...
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Prove $x^{46}+69x+2025$ is irreducible in $\mathbb Z[x]$
I was told to work in $\mathbb F_{23}$, and also show it has a linear factor $\mathbb Z_5$
Write $f(x)=x^{46}+69x+2025$. We begin by supposing that $f=gh$ for some $g,h \in \mathbb Z[x]$. First, $g$ ...
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Conceptual Explanation for Gauss Periods Normal Basis Criterion
I am studying the construction of normal bases using Gauss periods. Let $r = nk + 1$ be prime, $q$ a prime power with $\gcd(q, r) = 1$, and $\mathcal{K}$ the unique subgroup of $\mathbb{Z}_r^\times$ ...
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How to show that $\operatorname{lim}_{n \in \mathbb{N}} R[x_0, \ldots, x_n] \cong \operatorname{lim}_{(A, p(t))} A$?
Let $R$ be a commutative ring with unity. Let $I$ be the following index category:
The objects of $I$ are pairs $(A, p(t))$ where $A$ is an $R$-algebra and $p(t) \in A[t]$ is a polynomial
A morphism ...
- 4,970
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0
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51
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Prove that constant times polynomial is zero, given product of two polynomial is zero. [duplicate]
Problem statement: Suppose $R$ is a commutative ring, and $f$ is a nonzero polynomial in $R[x]$. Suppose there exists $g$, another nonzero polynomial in $R[x]$, and $f\cdot g=0$, prove that there ...
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Associated polynomial function vs evaluation map
In Bosch's "Algebra: From the Viewpoint of Galois Theory" (page 29), the author considers a ring extension $R\subset R'$, a polynomial $f=\sum_ia_iX^i\in R[X]$, and says that we can ...
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