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Questions tagged [algebras]

For questions about algebras, their properties, and their structures.

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How are these Quaternion Algebras isomorphic? $\Big(\dfrac{\mathit{a,b}}{\mathit{F}}\Big) \simeq \Big(\dfrac{\mathit{b,a}}{\mathit{F}}\Big)$

So let $\Big(\dfrac{\mathit{a,b}}{\mathit{F}}\Big)$ be a quaternion Algebra over a field $F$ with char $\neq 2$, and let $i,j$ be the standard generators for the quat. Algebra, meaning $i^2=a$ and $j^...
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About irreducible representations over the polynomial ring $k[x]$

From Example 2.3.14 (2) here page 21: Let $A = k[x]$. Since this algebra is commutative, the irreducible representations of $A$ are its 1-dimensional representations. They are defined by a single ...
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14 views

Symmetric Algebra over Tensor Product

Let $k$ be a field. I am interested in the symmetric algebra functor $S : k \text{-vect} \rightarrow k \text{-alg}$ taking a $k$-vector space $V$ to the symmetric algebra $S(V)$ over $V$, which is a ...
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1answer
35 views

Ring/Group actions on the definition of $R-$ modules and $R-$algebras.

Let $R$ be a commutative ring with $1_R$. I found in this post the definition of the action of a group acting on a ring. And then, the following questions came in my mind. Question 1. What is the ...
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18 views

Deducing the value of one parameter

I'm working with Javascript and I need some help because I'm not great at math. I have the following code: ...
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46 views

Show that $F[G]$ is isomorphic to $F[x]/(x^n - 1)$

Let me start by saying I know that there are similar problems on here but none that give any real sense of direction or understanding, at least for me. Let $F$ be a field such that $\text{char} (F)=p ...
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14 views

Proving that the set of all finite disjoint unions of mutually disjoint sets in a Semi Algebra, is the Algebra generated from the Semi Algebra

I am reading this book called as A probability path, by Sidney I Resnick. In this book he states the following axiom: Axiom And he goes on to prove it as follows: Proof Page 1 Proof Page 2 Can ...
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3answers
70 views

Embedding of $\mathfrak{sl}(2,\mathbb{C})$ in matrix algebra

Consider the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$ defined in terms of generators and relations by $$\langle x,y,h: [h,x]=2x, [h,y]=-2y, [x,y]=h\rangle.$$ If $X,Y,H$ denote the standard $2\times ...
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1answer
95 views

Proving that the first Weyl Algebra is simple

The first Weyl algebra over the complex numbers $\mathbb{C}[x]$ is defined to be the set $A_1 = {\{\sum_{i = 0}^{n} f_i(x) \delta^i : f_i(x) \in \mathbb{C}[x] }\}$. So it is the set of all linear ...
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Is the property of being contained in an affine domain preserved by quotients (of prime ideals)?

Let $K$ be a field and let $A$ be a $K$-algebra. Suppose $A$ is contained in an affine $K$-domain $B$ (that is a finitely generated $K$-algebra that is an affine domain) and let $P \in \operatorname{...
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1answer
37 views

What are some simple examples of algebras? [closed]

So an algebra $A$ over a field $K$ is a ring under the operations $+$ and $x$ and also a vector space under a scalar operation and the operation $+$. What are some examples of algebras? When rings ...
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1answer
62 views

Confusion regarding the definition of the Weyl Algebra

I've been studying algebraic structures recently; rings, fields, vector spaces and so on. I've recently just started learning what an algebra is, which, from what I can tell is a ring-like structure ...
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1answer
173 views

Ideals of a two-dimensional algebra with a given basis

My task is as follows: Find all ideals in a two-dimensional algebra $A$ over $\mathbb{R}$ with basis 1, $e$ where 1 is the multiplicative identity and Case 1: $e^2=0$, Case 2: $e^2=1$. My ...
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1answer
22 views

What is this module endomorphism for a given algebra?

I am having a little trouble understanding the following construction. Suppose $R$ is a commutative ring. Let $B$ be a unital associative $R$-algebra, and let $\{b_{i}\}_{i \in I}$ be a fixed ...
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1answer
61 views

Injective homomorphism $H:C_b(X) \to C_b(Y)$ implies the existence of a continuous and surjective map $F:Y \to X$

Let $X$ and $Y$ be $2$ topological spaces and let $C_b(X)$ and $C_b(Y)$ denote the set of all continuous and bounded functions on X and Y, respectively, to the space of complex numbers. It is a well-...
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Exercise on block theory of finite dimensional algebras

I am finding some problems in solving this exercise. Assume $A$ is a finite dimensional $K$-algebra, with $K$ an algebraically closed field. Let $B_1,\dots, B_r$ be the blocks of $A$, with unit ...
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Transpositions and idempotents in matrix algebras

Let $A$ be a semi-simple algebra over $\mathbb{C}$. An idempotent $x$ in $A$ is an element of $A$ such that $x^2 = x$. A minimal non-zero idempotent $y$ of $A$ is a non-zero idempotent of $A$ such ...
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1answer
67 views

Why doesn't this work for showing that $A[x]$ is a flat $A$-algebra?

$N$ is flat if and only if when $f: M' \rightarrow M$ is injective then $f \otimes \text{id}_N : M' \otimes N \rightarrow M \otimes N$ is injective ($N,M,M'$ are all $A$-modules). So consider $f: M' \...
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Checking a solution to a problem about non-zero divisors in an associative algebra having multiplicative inverses [duplicate]

The following problem appeared on a problem set I'm working on: "Let $A$ be a finite-dimensional associative algebra with $1$ over a field $k$. Show that an element $a$ in $A$ has a multiplicative ...
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1answer
43 views

On Wedderburn's theorem for $\mathbb{k}$-algebras of finite dimension with $\mathbb{k}$ algebraically closed.

I'm currently taking a first introduction to abstract algebra. At the moment, we're talking about semisimple modules and rings. We have already covered the following results Theorem (Wedderburn): $...
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1answer
25 views

A question on the product of integral homomorphisms

From the following question: Let $R,S_1,...,S_n$ be commutative rings, and suppose that $f_i: R\to S_i$ is an integral ring homomorphism for each $i$. Show that ring homomorphism $f:R\to \prod_{i=1}...
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Endomorphism of algebras checked on the generators

Say $A$ is an algebra with generating set $\{a_i\}_{i\in I}$, possibly infinite. I want to check that for my specific algebra, that some map $f:A\to A$ is an endomorphism. Can one check this on ...
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3answers
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Show that $\Bbb Q$ is not a finitely generated $\Bbb Z$-algebra.

Show that $\Bbb Q$ is not a finitely generated $\Bbb Z$-algebra. I know that $\Bbb Q$ is not a finitely generated $\Bbb Z$-module. From here how can I conclude that $\Bbb Q$ is not a finitely ...
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0answers
27 views

Algebra-module over a set, is there such a thing?

I'm reading a paper and $H$ is an algebra and $M$ is a set. And then they define $M|H$ to be the "$H$-module $M$". Is it obvious what this is or is it bad math description? My guess is the free $H$-...
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3answers
275 views

how to work out the values of integers in specific positions in a number

Apologies if this question is a duplicate, but I believe it is not. If there are three sets of numbers, A, B, and C, and each are integers $1\le n \le9$, occupying the hundreds, tens and unit ...
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1answer
31 views

Central simple quaternion algebra: why is the matrix for $\rho(v)$ antidiagonal?

Let $F$ be a field of characteristic $0$. Let $D$ be a central, simple quaternion division algebra over $F$. Let $x \in D$, not in $F$. Then $K = F[x]$ is a field of degree two over $F$, and $D$ is ...
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1answer
26 views

Relations of structures related to conjugate idempotents

If $e$ and $f$ are conjugate idempotents in some algebra $A$, I guess the modules $Ae$ and $Af$ should be isomorphic, as well as the algebras $eAe$ and $fAf$ . Are the maps canonically given by just ...
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Composition series of $\mathbb{P}_n $ as $\mathbb{R}[X]$ module.

Let $\mathbb{P}_n $ be the vectorial space of polynomials of degree $\leq n$. Let $T:\mathbb{P}_n \longrightarrow \mathbb{P}_n$ be the linear map given by $p(X)\mapsto p'(X)$. Calculate a composition ...
2
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1answer
41 views

Prove that $K \otimes_F F(\theta) \cong K[x]/(p(x))$ as $K$-algebras.

Let $F \subseteq K \subseteq L$ and let $\theta \in L$ with $p(x) = m_{\theta, F}(x).$ Prove that $K \otimes_F F(\theta) \cong K[x]/(p(x))$ as $K$-algebras. I've been trying to work through chapters ...
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2answers
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Associative algebra without nilpotent ideals is direct sum of minimal left ideals

In the book 'Spinors, Clifford and Cayley Algebras' Hermann states that any (finite dimensional) semisimple associative algebra is the direct sum of minimal left ideals. Here, 'semisimple' is defined ...
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How to determine what algebra I'm working with? [closed]

Suppose I have some mathematical object and can explore its properties. What is efficient way to determine which algebra this object corresponds to? (I'm not well acquainted with different kinds of ...
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1answer
31 views

Miscellaneous Questions involving Tensor Products, Exact Sequences, and Algebras

Let R be a ring (unital, though not necessarily commutative, and let: \begin{array}{ccccccccc} 0 & \hookrightarrow & M & \overset{f}{\hookrightarrow} & N & \overset{g}{\...
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1answer
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Why is $e \in C$ (a commutative subalgebra of $A$)?

I am reading the following proposition: Here $e$ represents the identity element of $A$ and $\sigma_C(x)$ and $\sigma_A(x)$ denote the spectrum of an element $x$ in $C$ and $A$ respectively. ...
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How to get an isomorphic ideal from a given one?

It's an open question, in general given an ideal in rings, algebras, lie algebras, etc., do you know how to get another one isomorphic to the given one? If you want to know the context: I have $I\...
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1answer
55 views

On the definition of free algebra and localization of a non-commutative ring

Below is a construction (p15) of the localization of a non-commutative ring $A$ by a subset $S$. Construction 3.1 Form the free algebra on a set which is in bijection with $S$ $$A\langle i_s \...
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28 views

Is multiplication in a normed algebra distributive?

I am reading C*-algebras and Operator Theory by Gerard Murphy. It defines an algebra as a vector space $A$ together with a bilinear map $$A^2\to A,\;\;\;(a,b)\mapsto ab,$$ such that $$a(bc)=(ab)c\;\;\;...
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1answer
48 views

Pushout in Commutative $\mathbb{Z}$-Alg

Note: Everything is commutative and unital I am asked to determine the pushout in the category of $\mathbb{Z}$-algebras. So far, I have shown that the pushout is a tensor product modded by the ...
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2answers
56 views

Question about R-algebra “generated by”

Background: In the context of Atiyah-Macdonald (so all rings are considered commutative and unital) an $R$-algebra, $A$, is defined to be a ring $A$ with a homomorphism $f\colon R \to A$. As ...
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0answers
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Spectral radius inequality for non-abelian Banach algebras

Background (question below) Let $A$ be a Banach algebra, i.e. $A$ is a complex vector space with norm $\left\Vert \cdot\right\Vert$ and multiplication satisfying $\left\Vert ab\right\Vert \leq\left\...
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1answer
43 views

Special $k$-algebra is finitely generated

Let $B$ be a $k$-algebra such that there exist finitely many elements $b_1,\dots,b_n \in B$ satisfying: $(b_1,\dots,b_n) = B$ (equivalently, there are elements $c_1,\dots,c_n \in B$ such that $\sum_{...
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1answer
53 views

Finitely generated $k-$algebras of regular functions on an algebraic variety

I am reading Q. Liu's "Algebraic geometry and arithmetic curves". In the proof of Lemma 4.3. at page 61 (the closed points of an open subset $U$ of an algebraic $k-$variety $X$ are closed in $X$), he ...
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1answer
124 views

All maximal subfields of a Division Algebra are isomorphic.

How can I show that two maximal subfields of a division-algebra have the same dimension over k. I found a simple proof here on page 4:http://www.math.northwestern.edu/~len/d70/chap17.pdf. Is there a ...
2
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1answer
40 views

idempotent in quiver theory

I am studyng quivers and algebras of the text "Elements of the representation theory of associative algebras", author Assem. In the page 46 says: "any idempotent $ε$ of $ε_a(KQ)ε_a$ can be written in ...
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3answers
36 views

Algebraic step including finite sum and binomial coefficient

Hello, I've stumbled into the following algebraic step in my combinatorics text book. Beside calculating it directly, I can't find a proper justification for it. Someone's got an idea?
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1answer
85 views

Module over finite dimensional simple algebra isomorphic to direct sum of minimal Ideal.

How can i show that a Left-Module M over a finte dimensional simple K-algebra A, that has finite dimension over K, is isomorphic to a direct sum of an minimal left ideal in A and the number of ...
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1answer
65 views

Reference Quest for Skolem-Noether-Theorem

What are good references for the Skolem-Noether theorem. I have to write about it so i would like to read a lot of proofs of this theorem. I only had two linear and two abstract algebra courses.
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Searching for Proofs of Wedderburn Theorem for finite dimensional simple Algebras

Hi im searching for proofs of the Wedderburn Theorem for finite-dimensional simple Algebras over a field. I know that there are a lot of proofs of Wedderburns Theorem for finite Divison-Algebras. Im ...
2
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1answer
60 views

Explicit concrete examples of k-affinoid algebras

I am having some troubles understanding $k$-affinoid algebras (where ($k$, |.|) is a complete, non Archimedean field, |.| is not trivial) and i am looking for some more concrete and particular ...
2
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0answers
90 views

Tensor product with an algebra bimodule

I am having a little difficulty in understanding the module structure of tensor products of modules over algebras (as opposed to arbitrary rings). To help me understand, here is an example. Let $K[x]$...
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1answer
75 views

How obviously injective is this “graded symmetrizer” map $\operatorname{S}(V) \to \operatorname{T}(V)$?

Starting with a graded vector space $V$, you can construct the tensor algebra $\operatorname{T}(V) := \bigoplus_{n>0} V^{\otimes n}$ and you can construct the symmetric algebra $\operatorname{S}(V) ...