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Questions tagged [algebras]

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

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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
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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
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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
118 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
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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
58 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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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
36 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
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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
60 views

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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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
274 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
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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
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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 ...
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1answer
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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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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
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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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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
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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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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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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
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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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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
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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
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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
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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 ...
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1answer
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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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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
67 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
59 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 ...
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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 ...
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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
72 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) ...
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2answers
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Descomposition of K-algebra

An element $e$ in a K-algebra $A$ is called an idempotent if $e^2 = e$. The idempotents $e_1, e_2$ ∈ $A$ are called orthogonal if $e_1e_2 = e_2e_1 = 0$. The idempotent $e$ is said to be primitive if $...
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Existence of submultiplicative norm on a $\mathbb{C}$-vector space

This is probably a very simple and maybe elementary question but, suppose we have a finite extension of $\mathbb{C}$ say $L$, is there always a sub-multiplicative norm on $L$, seen here as a finite $\...
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Isomorphism classes of algebras of rank 4 over an algebraically closed field

This work is done over an algebraically closed field $k$ (it can be assumed that $char(k)=0$). Now I consider all possible $k$-algebras $A$, up to isomorphism, with $rank_k(A)=4$ (i.e. $A$ has ...
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1answer
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Homomorphism Between 2x2 matrices and the Reals

Let $A=\begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}$. Let $\mathfrak{A}_\mathbb{R}$ be the real unital subalgebra of $M_2(\mathbb{R})$ generated by the matrix $A$. Prove that there are no ...
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Equivalence Classes and Quotient Algebras

I'm reading through this book: Algebras of Linear Transformations. Here is an image of a short passage: . I'm struggling to show that the multiplication of equivalence classes ([$a_1$][$a_2$] = [$a_1 ...
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2answers
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Definition of separable extension via $k$-algebra homomorphisms

I came across with a definition of separable field extension $F/k$ as below : Let $F/k$ be any finite extension. The extension is said to be separable if there exist $[F:k]$ distinct homomorphisms ...
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1answer
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Incidence algebras and dot products

Central question: since an arbitrary poset (or lattice) is not necessarily (comprised within) a vector space, how does one think about the structural similarity of convolutions on incidence algebras ...
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Leibniz rule and Alexander-Whitney coproduct

Is there anything more than a superficial similarity between the following? The Alexander-Whitney coproduct $\Delta$ on the tensor algebra $\bigotimes^\bullet V$ of a vector space $V$ is defined by ...