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Questions tagged [algebras]

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

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Equivalence of module structure involving an $A$-algebra

I'm new to modules and I'm trying to see the following equivalence: "If $A$ is a commutative ring, $R$ an $A$-algebra (i.e. a ring $R$ with a homomorphism $i : A \rightarrow Z(R)$) and $M$ an Abelian ...
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25 views

Computing Ext functor when no projective modules are known.

I have a $p^3$-dimensional (not semisimple) algebra $\mathcal{U}_{q}(sl_2)$ over $\mathbb{C}$ and i know how all its simple modules look like (there are $p$ of them, each $M_i$ has dimension $i$ for $...
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Every finite dimensional algebra is subalgebra of a matrix algebra

I have two questions about the following exercise: Let $K$ be a field and $R$ a $K$-algebra. Assume that $d := dim_K R$ is finite. Show that there is an injective $K$-algebra homomorphism $...
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1answer
67 views

Does this change of basis exist?

Let $A$ be a unital associative $\mathbb{C}$ algebra and let $M$ be a 2-dimensional $A$-module with basis $\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\}$. $A$ has two generators $\left\{a_{1}, a_{2}\...
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1answer
125 views

What's in a Noetherian $\mathbb{A}$-Module Ephemeralization?

Just kidding, it's not Noetherian. And "Emphemeralization" implies it is a physical construct, or that if it is, due to knowledge heretofore unbeknownst but recently gained by visualization of black ...
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49 views

k-algebra homomorphisms

I have the following question and I don´t know how to answer it: Let $k$ be an algebraically closed field (or just an arbitrary field), and $A$ and $B$ two $k$-algebras. Let $A$ be a $B$-module too....
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1answer
17 views

Given a $k$ algebra $A_0$, find another $k$ algebra $A$ with identity such that the extension $A_0\subseteq A$ has dimension $1$.

Let $A_0$ be a finite dimensional $k$-algebra where $k$ is an algebraically closed field of characteristic $p$ not necessarily with unit. We wish to show there is a $k$-algebra $A$ with a unit $1$ ...
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25 views

Product of open maps open? (not the cartesian)

Let $A$ be a locally convex algebra, or even just a topological algebra, and let $U_1,U_2\in A$ be open, is the product $$ U_1\cdot U_2=\left\{ a\cdot b\mid a\in U_{ 1} ,b\in U_{ 2} \right\} $$ ...
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41 views

Associated Graded Algebra

I'm trying to work through Exercise III.27 of Lang's Algebra: Let $A$ be a filtered algebra, $A=\bigcup_{j\geq 0}A_{j}$. For $j\geq 0$, define $R_{j}=A_{j}/A_{j-1}$, with $A_{-1}=\{0\}$. Let $R=\...
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Showing $M_1$ and $M_2$ are submodules of $M$, where $M$ is a module of a direct product algebra.

Let $A = A_1 \times A_2$, the product of two $k$-algebras. Suppose $M$ is some A-module. Define $M_1 \triangleq \{(A_1,0)m : m \in M\},\ M_2 \triangleq \{(0,A_2)m : m \in M \}$. Show that $M_1$ ...
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Let $k$ a field, $k'$ a subfield, and $A$ any associative $k$-algebra. Can a quotient of $A$ ever yield $k'$?

I am trying to learn some basic Scheme theory out of Eisenbud's book "Schemes: the Language of Modern Algebraic Geometry." I'm trying to understand how elements of a ring can be treated as functions ...
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16 views

Homomorphisms between matrix algebras

I was wondering about the following: Let $n,m,p\geq 1$ and let $\varphi\colon M_n(\Bbb C)\oplus M_m(\Bbb C)\to M_p(\Bbb C)$ be a homomorphism of $\Bbb C$-algebras such that $\varphi$ is injective ...
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Constructing the Algebraic Dual Space as a K-Algebra

Fix an arbitrary Field $K$ and suppose we are given a Vector Space $V\in{Obj(Vect_{K})}$, let me denote by $V^*:=Hom_{Vect_{K}}(V,K)$ the Algebraic Dual Space of $V$. It's clear that $[V\in{Obj(Vect_{...
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1answer
53 views

If B is a subalgebra of A, conclude that B closure is a subalgebra of A

If B is a subalgebra of A, conclude that $\bar{B}$ is a subalgebra of A. This is from Real Analysis by N. L Carothers chapter 12 exercise 3. The purpose of this is to lead up to the Stone Weierstrass ...
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1answer
38 views

Finitely generated $k$-algebras beginner examples.

I just found out about finitely generated $k$-algebras (where $k$ is a field). So it is an algebra $A$ for which we have a finite set of elements $(a_1,...,a_n)$ such that every element in $A$ can be ...
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99 views

Finite Algebras and Grobner Bases

Background Suppose that $A$ is a finite $\mathbb{R}$-algebra, that is, it is finite-dimensional as an vector space. By a consequence of the Hilbert-basisatz, since $\mathbb{R}$ is Noetherian, then so ...
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23 views

Unique multiplicative quadratic form on quaternion algebras

I want to prove, that the only multiplicative quadratic form $Q$ (so $Q(xy)=Q(x)Q(y) \forall x,y$) on a quaternion algebra $\Big(\dfrac{a,b}{F}\Big)$ is the norm $\mathrm{Nr}$, which is isometric to ...
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43 views

Equivalent definitions of Clifford algebra, verification

Let $(V,B)$ be a finite dimensional $k$ vector space $V$ with an associated quadratic form $Q$. $char \, k \not= 2$. Let $X:= \{e_i \}_{i=1}^n$ be a set of basis for $V$. Construct $k\langle X \...
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1answer
30 views

Representing an finite-dimensional algebra over a field using a quotient ring

My professor was talking about different ways to think about an algebra $A$ over a field $k$. One that she mentioned briefly but didn't go into much detail on was that of a quotient ring. Roughly the ...
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1answer
34 views

Representing an algebra over a field using a quotient ring

My professor was talking about different ways to think about an algebra $A$ over a field $k$. One that she mentioned briefly but didn't go into much detail on was that of a quotient ring. Roughly the ...
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1answer
47 views

Classification of subalgebras of composition algebras

Let $F$ be an algebraically closed field. It is known that the only composition algebras over $F$ are $F$ itself, the direct sum $F\oplus F$ (also called split-complexes), the algebra of $2\times 2$ ...
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1answer
29 views

Finding the Spectrum of an element of $\ell^\infty$

I have done Q.1.2.1 already and it is quite clear.But I am not sure about the next one. How does the closure come into the picture. I have a feeling that it should be $f(S)$ only. Am I missing ...
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102 views

quasi isomorphism of two dg algebras

I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is ...
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2answers
58 views

What is the Gelfand-Naimark representation of functions that don't vanish at infinity?

The Gelfand-Naimark theorem says that every commutative C*-algebra is isometrically isomorphic to $C_0(X)$, the set of continuous functions $f:X\rightarrow\mathbb{C}$ that vanish at infinity, for some ...
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177 views

No onto map between group algebras $FS_5$ onto $M_6(F)$.

I have to prove that there does not exist a surjective group algebra homomorphism from $FS_5$(the group algebra of the symmetric grpoup, $S_5$, over the field $F$) to $M_6(F)$, where $F$ is the field $...
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42 views

Tensor product of subalgebra

Let $R$ be a ring and $A, B, C$ $R$-algebras such that there is an injective $R$-algebra Homomorphism $i: B \hookrightarrow C$. Is it true that the induced map $j: A \otimes_R B \to A \otimes_R C$ is ...
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1answer
49 views

The category of $T$ algebras on Set is equivalent to the category of monoids

Let Set denote the category of sets. Let $T:$ Set $\to$ Set be the functor that sends a set $X$ to the set of finite words on $X$. That is, $TX = \{[x_m,..,x_1] : m = 0,1,2,3..., x_i \in X\}$ $T$ ...
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1answer
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The correspondence between maximal ideals in an algebra and it's unitalization

Let $A_+$ denote the unitalization of a $\mathbb{C}$-algebra $A$ ( which is $A \oplus \mathbb{C}$ endowed with well-know multiplication rule. I know that the map $\Omega(A_+) \to \Omega(A)$, $J \...
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1answer
65 views

On proposition I.1.2 of “Quantum Groups” by Christian Kassel

I am working through Christian Kassel's textbook on Quantum Groups. The Proposition states that 5 statemens are equivalent. The two I am having trouble with are as follows. 1.For any pair $V'\subset ...
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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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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
40 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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54 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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24 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
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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
169 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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39 views

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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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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2answers
210 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
26 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
70 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
73 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
47 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
27 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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0answers
26 views

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 ...