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Questions tagged [algebras]

For questions about algebras, their properties, and their structures. Use [tag:algebra-precalculus] or [tag:abstract-algebra] if your question is about algebra, not algebras.

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Bilinear form on a finite dimensional algebra over a field

I am reading the first chapter of 'methods of representation theory, Volume I' section 9A, written by Charles W.Curtis & Irving Reiner. Let $A$ denotes a finite dimensional algebra over an ...
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1 vote
1 answer
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$\operatorname{End}_S(M \otimes_R S) \cong \operatorname{End}_R(M) \otimes_R S$?

Let $R$ and $S$ be commutative rings with unity. Let $f: R \to S$ be a ring homomorphism. Let $M$ be an $R$-module. Do we have $$ \operatorname{End}_S(M \otimes_R S) \cong \operatorname{End}_R(M) \...
Smiley1000's user avatar
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1 vote
1 answer
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Bialgebra structure on self dual associative algebra

Given a finite-dimensional associative $\mathbb{k}$-algebra $A$, one can define its dual $A^*$ as the $\mathbb{k}$-vector-space $\operatorname{Hom}_{\mathbb{k}}(A, \mathbb{k})$ with $A$-multiplication ...
Jannik Pitt's user avatar
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3 votes
1 answer
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Is the set $C^{n}([a, b])$ a normed algebra with the pointwise product and the sum of suprema norm?

Given the pointwise product $\cdot$ in $C^{n}([a, b])$, and the norm $\lVert \hspace{0.2 cm} \rVert$ defined for any $n$-times continuosly differentiable function $h$ whose domain is $[a, b]$ as ...
Emilio Mora's user avatar
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60 views

$A\otimes_kB$ is an integral domain when $k$ is algebraically closed, and $A$ and $B$ are finitely generated

I am trying to understand this answer in this math stack exchange post I don't see how it works. Let $A$ and $B$ be finitely generated $k$ algebras with $k$ algebraically closed, and suppose that $A$ ...
Chris's user avatar
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1 vote
1 answer
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In this proposition why is $\approx$ used rather than $=$?

Proposition 1.1.3 Residuated Structures in Algebra and Logic by Metcalfe, Paoli, and Tsinakis states in part: There exists a bijective correspondence between lattices and algebras $\langle L, \wedge, \...
Jay's user avatar
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1 vote
0 answers
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Determining initial algebras and final coalgebras for a given functor without using limits/colimits

I'm trying to find the final coalgebra for a certain functor but I have no idea how to do that in general, so I was hoping to go through the process with some simpler examples. In section 4.1, The ...
msb15's user avatar
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1 vote
2 answers
30 views

Find three finite dimensional semisimple $\mathbb{R}$ algebras

I'm asked to find three finite dimensional semisimple $\mathbb{R}$-algebras, $R$, each with only one simple module $S$ up to isomorphism so $R\simeq S\oplus S$ as $R$-modules. I'm guessing I need to ...
quanticbolt's user avatar
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1 answer
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Artin-Wedderburn application - finding simple modules

I'm trying to solve the following: Let $R$ be the $\mathbb{R}$-algebra $R:=M_2(\mathbb{R})\times M_3(\mathbb{H})\times\mathbb{C}\times\mathbb{C}$. Determine how many simple (left) $R$-modules there ...
quanticbolt's user avatar
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1 answer
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Characterizing congruences on the algebra of natural numbers

I'm trying to do Exercise 31 in Jan Rutten's book on coalgebras. The goal is to show that, given a characterization of congruences on the initial $N$-algebra $(\mathbb{N},[\text{zero},\text{succ}])$, ...
msb15's user avatar
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3 votes
1 answer
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How can you tell if an exponential function has undergone transformations from a table?

Here's a scenario. Say there are three points on a table: $(0,1.5)$, $(1,3)$, and $(2,6)$. The equation for the exponential function modeled by these points is $y=a\times b^{x-1}$. However, after ...
user386598's user avatar
0 votes
1 answer
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Endomorphisms of $\mathbb{C}^2$ as an associative unital $\mathbb{C}$-algebra

Given a commutative unital ring $R$ and associative unital $R$-algebras $A$ and $B$, call a map $\varphi: A \to B$ a morphism of associative unital $R$-algebras if it satisfies the following ...
Smiley1000's user avatar
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1 vote
1 answer
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Confusion about a result in Pierce's book Associative Algebras.

I am confused about two Lemmas in the book Associative Algebras by R.Pierce (pages 43-44). In Lemma d, since each right ideal of a can be considered an algebra over $A$ then if $N$ is a minimal right ...
Adam's user avatar
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2 votes
1 answer
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Is there a good presentation of the matrix algebras?

Let $R$ be a commutative ring with identity. Suppose the matrix algebra $\operatorname{Mat}_n (R)$ and matrices $$E_{mk}:=(a_{ij})=\begin{cases} a_{ij}=1 \ \text{if} \ i=m \ \text{and} \ j=k \\ a_{ij}=...
Mitya Kustov's user avatar
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spectrum of a general algebra

For unital Algebras $A$, the Spectrum of an element $x \in A$ is defined to be $$ Sp_A(x) := \{ \lambda \in \mathbb{C}: x - \lambda * e \text{ is not invertible} \} $$ According to my textbook, there ...
Olimani's user avatar
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1 answer
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Zero divisors in an algebra with two generators [closed]

Let $k$ be a field, and $R = k\langle x,y \mid x^2 = 0\rangle$. The elements $x$ and $y$ are not supposed to commute with each other. Is the only case where nonzero elements $a, b \in R$ satisfy $ab=0$...
Ralle's user avatar
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0 answers
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Generalization of Cartesian Product with Shared Actions for Reactive Graphs

The defnitions of Reactive Graph are the follow: Multi-Action Reactive Graph of order $n$ is a structure $M = (w, \mathcal{F} ,\mathcal{A})$ where: $w \in Q $ is the current state; $\mathcal{F} = (Q,...
Dtinas10's user avatar
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1 answer
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matrix with univariate entries: rank deficit of specialization ≤ vanishing order of determinant, part II: commutative ring

a special case of this question with coefficients in a field was recently asked and answered. fix a commutative ring with unit $R$, and an $n \times n$ matrix $M(X)$ with entries in $R[X]$. the ...
BD107's user avatar
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0 answers
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divisibility in polynomial ring: if $(X - a)^t | f(X)$ and $(X - b)^t | f(X)$, then does $(X - a)^t \cdot (X - b)^t \mid f(X)$? [duplicate]

let $K$ be a field an $A$ a commutative $K$-algebra with unit. fix an element $f(X) \in A[X]$, unequal elements $a$ and $b$ of $K \subset A$, and an integer $t > 0$. Question. if $(X - a)^t \mid f(...
BD107's user avatar
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5 votes
1 answer
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Show that $\mathbb Q[t] \otimes_{\mathbb Q[t^2]} \mathbb Q[t] \cong \mathbb Q[x,y]/(x^2-y^2)$ (as $\mathbb Q$--algebras)

Show that $\mathbb Q[t] \otimes_{\mathbb Q[t^2]} \mathbb Q[t] \cong \mathbb Q[x,y]/(x^2-y^2)$ (as $\mathbb Q$--algebras) So I first tried to show this by first defining the map $\varphi: \mathbb Q[t] ...
Squirrel-Power's user avatar
1 vote
1 answer
52 views

Category of monoids isomorphic to coslice category

Let $k$ be a commutative ring and $R$ a $k$-algebra. The category $R\text{-Bimod}$ of $R$-bimodules becomes a monoidal category with the tensor product of $R$-bimodules. Denote the category of $k$-...
Margaret's user avatar
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0 votes
0 answers
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Sufficient conditions for unital sub-algebra of matrix ring to be closed under inversion

Let $n \ge 1$ and R be a unital $\mathbb{R}$-subalgebra of $M_n(\mathbb{R})$. If $A \in R \cap GL_n(\mathbb{R})$, is there any criteria to guarantee that its inverse is also in R? Since R is a vector ...
Pedro Lourenço's user avatar
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0 answers
78 views

$\mathbb{R}$-Algebra isomorphism $\mathbb{C}[z;\sigma]\cong \mathbb{R} \langle x,y\rangle/(y^2+1,xy+yx)$

Considering the algebra $A=\mathbb{C}[z;\sigma]$ of the skew polynomials, which is like $\mathbb{C}[z]$ but with the multiplication of elements defined the following way: $xb=\sigma(b)x$ and extending ...
Alex A.G.'s user avatar
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What are some good learning materials for Affine Algebras

I am looking for resources - courses or books - that graduate students can take to learn Affine Algebras, preferably along with their generalizations and applications to physics. This post is inspired ...
Mahammad Yusifov's user avatar
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1 answer
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Reduced finitely generated k-algebras are isomorphic to $k^n$

We let $k$ denote a fixed algebraically closed field of characteristic zero. We let $R$ denote a reduced finitely generated $k$-algebra where $\dim_kR = n$ as a $k$ vector space and $n$ is a positive ...
Jeff's user avatar
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1 vote
0 answers
47 views

Reference needed: Category of modules over Frobenius algebra is a Frobenius category

Someone told me recently that the category of modules over a Frobenius algebra is a Frobenius category. Where can I find a reference for a proof of this? Since the category of modules must be an ...
user829347's user avatar
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1 vote
1 answer
80 views

Understanding the first Weyl algebra $A$ and $\operatorname{End}_A (V)$

Let $\mathbb{F}$ be a field, $V=\mathbb{F}[x]$ the vector space of polynomials. Suppose we have the first Weyl algebra $A$, the $\mathbb{F}$-subalgebra of $\operatorname{End}_{\mathbb{F}}V$ generated ...
Alex A.G.'s user avatar
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3 votes
1 answer
69 views

Question about If $B$ is a $A$-algebra and $N$ is a $B$-module, then can $N$ naturally become a $A$-module?

So I think the only point is to show how to build the $A$-module structure on $N.$ We know that $B$ is an $A$-algebra which is also an $A$-module where the $A$-module structure is given by the scalar ...
Beginner's user avatar
1 vote
1 answer
118 views

Why do we divide annual interest rate by number of compounding periods in the compound interest formula?

I understand that this is an extremely basic question and has been asked before, but we just learned this in school and as I am in eighth grade, I don't really understand previous explanations. I am ...
user386598's user avatar
1 vote
0 answers
86 views

Prove that $\mathbb{R}[x]/(x^2+1)^2$ is isomorphic to $\mathbb{C}[y]/(y^2)$ as a $\mathbb{C}$-algebra.

How could I prove that $A=\mathbb{R}[x]/(x^2+1)^2$ is isomorphic to $\mathbb{C}[y]/(y^2)$ as a $\mathbb{C}$ algebra? Are there any known isomorphisms or is there a trivial way to do so? I have been ...
Alex A.G.'s user avatar
  • 177
0 votes
1 answer
73 views

Equivalence of two algebras

I am a theoretical physicist and in the context of my research I find myself with two algebras which should both correspond to the $\mathfrak{sl}(2,\mathbb{R})$ Lie algebra, but I can't seem to find ...
LuVa's user avatar
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1 vote
1 answer
125 views

Small generating set for the unique minimal prime ideal of a finitely generated $\mathbb{C}$-algebra

Let $A$ be a $\mathbb{C}$-algebra, generated by $n$ elements ($n$ finite). Assume that $A$ has a unique minimal prime ideal $\mathfrak{p}$. Write $t$ for the minimal number of generators for the ideal ...
Object's user avatar
  • 339
2 votes
1 answer
204 views

Module structure on $R$-algebra $S$ by restriction of scalars and splitting of the algebra map

Let $R, S$ be commutative Noetherian rings. Let $f: R \to S$ be a ring homomorphism. Consider the $R$-module $f_* S$ whose underlying abelian group is $S$ itself but the $R$-module structure is given ...
uno's user avatar
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4 votes
2 answers
95 views

Does there exist an $\mathbb{R}$-algebra in which exponentials are especially well-behaved?

Question. Does there exist a finite-dimensional commutative $\mathbb{R}$-algebra $X$ equipped with a homeomorphism $$\exp : X \rightarrow X \setminus \{0\}$$ satisfying $\exp(0) = 1$ and $\exp(x+y) = \...
user avatar
1 vote
0 answers
43 views

A question about the minimal number of operations required to obtain a matrix with generic eigenvalues

Let $A,M \in \operatorname{Mat}_n(\mathbb{C})$ be $n \times n$ matrices such that $M$ is invertible and $MA \neq AM$. Consider the algebra $\mathcal{A}$ generated by the set $\{I,A,MAM^{-1}\}$, where $...
hugo's user avatar
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2 votes
1 answer
69 views

Homotopy of maps of $A_{\infty}$-algebras

Let $f, g: A \to B$ be maps of $A_{\infty}$-algebras. What is the correct (explicit) notion of a homotopy between $f$ and $g$? This is given in the expository paper of Keller in terms of maps of the ...
JD1874's user avatar
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1 vote
0 answers
61 views

On the homomorphism as $M_n(A)$-modules.

I'm trying to prove that, if $A$ is an Algebra over a field $F$, and $U,V$ are $A-$modules. Then any element in $Hom_{M_n(A)}(U^n, V^n)$ (recall that $U^n, V^n$ is the set of size $n$ vectors as an $...
Nestor Bravo's user avatar
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0 answers
26 views

Turning a subalgebra into an ideal by changing the algebra it is contained in

Let $B$ be an (not necessarily unital) algebra and $A \subseteq B$ a subalgebra. Is there some sort of quotient $C$ of $B$ (or any other interesting construction not necessarily found by taking ...
Francisco's user avatar
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1 vote
1 answer
79 views

Semisimplicity implies separability for a perfect field

Let $k$ be a field, $A$ a finite-dimensional semisimple $k$-algebra. If $k$ is a perfect field (every finite field extension of $k$ is seperable), then $A$ is separable. I know a proof that uses the ...
Margaret's user avatar
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3 votes
1 answer
57 views

Does there exist a topological power-associative non-metric unital algebra $V$ such that $\exp x$ is defined for all $x\in V$?

From what I understand, we normally define the operator $\exp$ only on unital Banach algebras, because the triangle inequality ensures that the infinite series converges: $$\lim_{n\to\infty}\left\|\...
Kyan Cheung's user avatar
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0 votes
0 answers
32 views

$\operatorname{Hom}_A(M, N)$ and $M^* \otimes_A N$

If $U$ and $V$ are vector spaces over some field $k$ and $U$ is finite dimensional with basis $\{u_i\}_i$, then we know that $\def\Hom{\operatorname{Hom}}$ $$\Hom_k(U, V) \to U^* \otimes_k V,\quad f \...
Bubaya's user avatar
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23 votes
6 answers
3k views

Proof the quaternions are 4-dimensional?

The quaternions can be defined as $$\mathbb{R}\langle X,Y\rangle/(X^2+1,Y^2+1,XY+YX)$$ From these relations, it is relatively easy to prove that $1,X,Y,XY$ span the quaternions over $\mathbb{R}$. But ...
Zoe Allen's user avatar
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0 votes
0 answers
21 views

Inferring classification of Clifford algebras from classification of Clifford modules

Let $Cl_n$ be the Clifford algebra (over reals) $$ Cl_n = T^{*}\mathbb{R}^n/\langle v\otimes v - q(v) \rangle. $$ There is a periodic table of $K$-representations of $Cl_n$, i.e. $\mathbb{R}$-linear ...
Hyeongmuk LIM's user avatar
3 votes
0 answers
122 views

Equivalent definitions of Hopf algebras

Recently, I started to study the book Hopf algebras by Moss Sweedler, in such book, given a coalgebra $(C,\Delta,\epsilon)$ and an algebra $(A,\mu,\eta)$, the autor defines the convolution of two ...
ferolimen's user avatar
  • 630
2 votes
1 answer
117 views

Jordan-Holder theorem for group algebras

I'm currently studying the Jordan-Holder theorem for modules and representations of associative algebras over fields. I was wondering if there is a way to prove the Jordan-Holder theorem for finite ...
Lorenzo Ferraiuolo's user avatar
2 votes
1 answer
68 views

Definition of splitting field: Why do we require centrality?

Let $k$ be a field. Let $D$ be a division algebra over $k$. Call a field extension $K/k$ a splitting field for $D$ if there exists a positive integer $n$ such that $D\otimes_k K\cong M(n\times n,K).$ ...
Margaret's user avatar
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0 votes
1 answer
91 views

Complete Subalgebras and Dense Subalgebras of Complete Boolean Algebras are Regular Subalgebras

In Thomas Jech's Set Theory book, he states that If $A$ is a complete subalgebra of a complete Boolean algebra $B$ then $A$ is a regular subalgebra of $B$. Also, if $A$ is a dense subalgebra of $B$ ...
Ali Dursun's user avatar
1 vote
1 answer
188 views

Finding the discriminant of a quaternion algebra

Consider the totally real number field $ F=\mathbb{Q}(\zeta_{10}+\zeta_{10}^*) $. Consider the quaternion algebra $ Q=(\frac{-1,-1}{F}) $. How do I compute the discriminant of this algebra? I gave ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
46 views

On the property of a ring modulo its Jacobson radical being a division ring

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $A$ be a finitely generated associative $R$-algebra. Let $x\in \mathfrak m$ be a non-zero-divisor on $A$ such that $xA\neq A$. If $A/J(...
uno's user avatar
  • 1,560
1 vote
0 answers
67 views

Definition of finite-dimensional 'self-dual algebra' over a field

In some informal notes that are not publicly available and for which I do not have permission to reproduce here, there is a reference to a 'finite-dimensional self-dual algebra over a field $K$'. I ...
user829347's user avatar
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