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Questions tagged [algebras]

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

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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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21 views

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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2answers
27 views

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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14 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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12 views

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
47 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
27 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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96 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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22 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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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
28 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
33 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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0answers
14 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
25 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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92 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
45 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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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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28 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
46 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
58 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
36 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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0answers
49 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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18 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
126 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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37 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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1answer
45 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
64 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
179 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
23 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
62 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
71 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
44 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
26 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
77 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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0answers
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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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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
30 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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14 views

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
48 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 ...
3
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2answers
46 views

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 ...