Questions tagged [algebras]

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

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votes
0answers
10 views

Modules over the dual of an infinite dimensional coalgebra

Let $k$ be a field and let $A$ be a finite dimensional (unital, associative, not necessarily commutative) $k$-algebra. The $k$-linear dual of $A$ is a coalgebra, and viceversa, the $k$-linear dual of ...
2
votes
0answers
50 views

Well-defined way to quotient by a relation involving an infinite sum

Take a unital (*-)algebra $\mathcal A$ generated by a finite set of generators, $e_n$, and relations. We can require the generators to be (self-adjoint) projections. Some of the relations are of the ...
1
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0answers
30 views

Showing $\lambda: A \otimes C^* \rightarrow \text{Hom}(C,A)$ is a morphism of algebras

Show that $\lambda: A \otimes C^* \rightarrow \text{Hom}(C,A)$ is a morphism of algebras. Let either $C^*$ or $A$ be finite dimensional, and let $\lambda$ be the isomorphism $\lambda: A \otimes C^* \...
1
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1answer
41 views

$\mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$ is an ideal of $\mathfrak{J}_{ii}$

Let $\mathfrak{J}$ be a Jordan algebra, and $\mathfrak{J} = \sum \mathfrak{J}_{ij}$ the Peirce decomposition of $\mathfrak{J}$ relative to orthogonal idempotents $e_i$ with sum $1$. Prove that $\...
17
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6answers
2k views

Are all finite-dimensional algebras of a fixed dimension over a field isomorphic to one another?

Suppose I have a finite-dimensional algebra $V$ of dimension $n$ over a field $\mathbb{F}$. Then $V$ is an $n$-dimensional vector space and comes equipped with a bilinear product $\phi : V \times V \...
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0answers
44 views

If $S$ is a free $R$-algebra, then no element of $S$ is integral over $R$? And other questions.

Suppose $R$ and $S$ are commutative rings. If $S$ is any $R$-algebra (not necessarily finitely generated), then every element of $S$ is written as a polynomial with coefficients in $R$ ? If $s \in S$...
2
votes
2answers
99 views

Term for an additive abelian group equipped with a multiplication which is distributive over the addition, but not necessarily associative?

What is the official name of an additive abelian group with a biadditive multiplication (left and right distributivity of multiplication over addition and no other assumptions)?
3
votes
0answers
191 views

Morphisms of rings that define morphisms of derivations

I am not super experienced in (commutative) algebra and in the course of some of my work I noticed that I had a need for morphisms of rings that are compatible with the asssociated module of ...
0
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1answer
41 views

The duality on projective modules takes minimal presentations to minimal presentations

I've been working through Auslander, Reiten, and Smalø's Representation Theory of Artin Algebras. In reading through the basic setup relevant to the transpose operation, I've come across two ...
3
votes
2answers
101 views

If the monoid algebra $R[M]$ is finitely generated, then $M$ is a finitely generated monoid.

Consider a commutative, cancellative, torsion-free monoid $M$ and a commutative ring $R.$ If the monoid algebra $R[M]$ is finitely generated as an $R$-algebra, then $M$ is finitely generated. I am ...
0
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0answers
38 views

Find an algebraic division algebra that is not finite dimensional

I want to find an algebraic division algebra that is not finite dimensional, but i don't want to do it in terms of field extensions nor anything like that. Instead of that, what i want to do is to ...
1
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0answers
29 views

Need help under standing the definition of the Weyl Algebra, $A_n$.

From the source I'm reading (S. C. Coutinho, A Primer of Algebraic D-Modules), $A_n$ is defined to be the subalgebra of $\text{End}(\mathbb{C}[x_1,..., x_n])$ generated by the operator $$x_i : \mathbb{...
0
votes
2answers
31 views

If $U,V$ are left $A$-modules, then is $U \otimes V$ a left $A \otimes A$-module?

Let $A$ be a unital associative algebra over the field $k$ and $U,V$ left $A$-modules. A book I'm reading claims that $U \otimes V$ is a left $A \otimes A$-module. First, how is the tensor product ...
0
votes
2answers
39 views

Associative algebras with commutative multiplication?

I.e. the bilinear map/product is not only associative, but also commutative. I am looking for examples of unital associative algebras, so they should be a vector space and a ring, not a vector space ...
1
vote
1answer
54 views

Trace and higher traces on an algebra

Let $A$ be a $k$-algebra for some field $k$. In Loday 2.4.5 he calls maps of the form $$\{f:A\to k|f(a_1a_2) = f(a_2a_1)\}$$ a trace on $A$, which he identifies as the $0$-th cyclic cohomology of $A$, ...
1
vote
0answers
17 views

Total number of nonisomorphic uniserial modules for a Nakayama algebra

Let $\Lambda$ be a Nakayama algebra and $\Lambda=\coprod\limits_{i=1}^sn_iP_i$ with $P_i$ nonisomorphic indecomposable projective modules ($n_iP_i$ denotes $n_i$ copies of $P_i$). Prove that the total ...
0
votes
1answer
25 views

For an algebra $A$, show that the bimodule $A \otimes_{A \otimes A^{\text{op}}} A \cong \frac{A}{[A,A]}$.

If A is an associative k-algebra, and $A^{\text{op}}$ represents the opposite k-algebra (i.e. $a*_{A^{\text{op}}} b := b\cdot a)$. In the following we consider ${\cal A_1}=A$ as a right $A\otimes A^{...
0
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0answers
16 views

Can every simple $A\otimes_k B$-module be realized as $L\otimes _k N$, where $L$ and $N$ are simple $A$ and $B$-modules, respectively?

Suppose $A$ and $B$ are finitely generated algebras over a field $k$. Let $M$ be a simple $A\otimes_k B$ module. Can one write $M\simeq L\otimes_k N$, where $L$ is a simple $A$-module, and $N$ is a ...
5
votes
0answers
53 views

Is there a name for algebras over a field $k$ whose residue class fields have finite dimension over $k$?

Let $k$ be a field and let $A$ be a $k$-algebra. Assume that for every maximal ideal $P \subseteq A$ the residue class field $A/P$ has finite dimension as a $k$-vector space. Is there a name for $...
5
votes
1answer
63 views

Cyclic cohomology of a field $k$

Let $A$ be a $k$-algebra where $k$ is a field. Define $C^n(A):=\text{Hom}_k(A^{\otimes n+1}, k)$, where $A^{\otimes n+1}$ is the $n$-fold $k$ tensor product of $A$ with itself. Then the cyclic ...
1
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0answers
43 views

Symmetrization map

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra, define a symmetrization map $S\mathfrak{g} \to U\mathfrak{g}$ by $x_1 .. x_n \to \frac{1}{n!} \sum x_{\sigma(1)} .. x_{\sigma(n)}$ (here $x_j$ ...
4
votes
2answers
51 views

If $A$ is a real or complex algebra and $a\in A$ is such that $ab=0$ for all $b\in A$, then $a=0$?

Let $A$ be an (not necessarily unital) algebra over $\mathbb{R}$ or $\mathbb{C}$. If $a\in A$ is an element such that $ab=0$ for all $b\in A$ (or equivalently, $aA=\{0\}$), can we then conclude that $...
1
vote
0answers
23 views

Dimension of the dual of a simple module over a simple $\mathbb{C}$-algebra.

I am given that A is a simple finite-dimensional associative unital algebra over the $\mathbb{C}$, and $M$ is a simple $A$-module. Furthermore, $V_M = \text{Hom}_A(M, A)$ is a right A-module with the ...
0
votes
0answers
15 views

Non-semi-simple finite-dimensional algebras

I am studying finite-dimensional algebras over a field (of zero characteristic) and am looking for examples of (unital) algebras which are not semi-simple. Of course, path algebras of quivers give a ...
1
vote
0answers
18 views

If two algebras are generated by the same elements, then are they isomorphic?

The $0$-Hecke algebra $\mathcal{H}_0(S_{n+1})$ is generated by $\{ h_1, \ldots,h_n\}$ satisfying i) $h_i^2=-h_i$ ii) $h_ih_j=h_jh_i$ if $|j-i| > 1$ iii) $h_ih_{i+1}h_i=h_{i+1}h_ih_{i+1}$. Now, ...
1
vote
1answer
27 views

Hom over algebras and modules

I'm working on the following question: Let $R$ be a ring (commutative and unital), $f: R \rightarrow B$ be a $R$-algebra, $M$ an $R$-module and $R$ a $B$-module. Show that $\text{Hom}_B (B \otimes_R ...
1
vote
1answer
16 views

Find all algebras with one binary operation and the set {0,1}

I believe the only algebra is (A,*). +,-,/ don't meet the requirements. But how do I prove that there is no other? What are all the binary operations on numbers?
2
votes
2answers
57 views

What's the distinction between an algebra and a Lie algebra?

An algebra, as far as I know, is closely related to a group with a family of functions being closed under addition, scalar multiplication and then the product of any two functions in the family. Then ...
1
vote
1answer
16 views

complex metric space, algebra that doesn't separate points, extension of baby Rudin

Let the metric space be $(C([-\pi, \pi]), \mathbb{C})$ with the uniform metric. Define $f_k=e^{ikx}$ where $-\pi\le x\le\pi$. Let $A$ be the algebra of $\{f_k\}$ where $k$ is a nonnegative integer, ...
1
vote
1answer
21 views

$A$-linear maps between $A$-modules where $A$ is a $K$-algebra and $K$ is a commutative ring

This is from Alexander Zimmermann's Represenation Theory. How can we talk about "$K$-linearity" of an $A$-module homomorphism $\alpha:M\to N$?
1
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0answers
46 views

Checking that $\dfrac{\mathbb{Z}[x_1,x_2]}{\langle x_1^2\rangle}$ is a $\mathbb{Z}$-algebra

From NPTEL's Commutative Algebra online course, lecture 17 (link here in the time of the video the example is given), it is given the following example concerning a non-finite $\mathbb{Z}$-algebra: ...
0
votes
0answers
39 views

Finding element such that $B/\mathfrak{m}$ is finitely generated as a module.

The general problem is this: Let A be a commutative ring, $f:A \to B$ a ring homomorphism making $B$ a finitely generated $A$-algebra, and let $\mathfrak{m}$ be a maximal ideal of $B$ lying over a ...
2
votes
1answer
50 views

Ideal not contained in maximal ideal in Banach Algebra $C_0(\mathbb{R})$

Let $A = C_0(\mathbb{R})$ be the (non unital) commutative Banach Algebra (with the uniform norm) of continuous functions vanishing at infinity. Let $$ I=\lbrace f \in A : \lim_{x \to \infty} xf(x) = ...
1
vote
1answer
56 views

How are the following two rings isomorphic?

The paper here in Construction 4.16 makes the following claim that I'm unable to unpack (though my question should be self-contained here): Let $R$ be a commutative ring, and let $r\in R$ be an ...
1
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0answers
36 views

What's an example of a weakly closed nil-subset whose enveloping algebra is not nilpotent?

Let $A$ be an associative $k$-Algebra. $W\subseteq A$ is said to be weakly closed if there is a $\gamma\colon W^2\to k$ such that $W$ is closed under the $γ$-bracket $[a,b]_γ=ab+γ(a,b)ba$. Jacobson ...
1
vote
1answer
42 views

Why are rings with identity $\mathbb{Z}$-algebras?

According to Dummit and Foote's definition, an $R$-algebra ($R$ is a commutative ring with identity) is a ring A with identity together with a ring homomorphism $f: R \rightarrow A$ mapping $I_R$ to $...
0
votes
1answer
45 views

Homomorphism between polynomial algebras

$K$ is a field and $K[X_1,\dots, X_n]$ and $K[X_1,\dots, X_m]$ are polynomial algebras in $n$ and $m$ indeterminates respectively. Suppose there is a $K$-algebra homomorphism $\phi:K[X_1,\dots, X_n] \...
3
votes
0answers
55 views

$ \mathcal{F}_n = \sigma(\{0\},\{1\},\{2\}, \dots , \{n\})$ showing $\cup_{n\geq0} \mathcal{F}_n $ is not a sigma algebra

Problem : We consider on $\mathbb{N}$, for all $n\geq 0$ the sigma algebra $ \mathcal{F}_n = \sigma(\{0\},\{1\},\{2\}, \dots , \{n\})$ show that the sequence of sigmas $(\mathcal{F}_n, n\geq 0)$ is ...
0
votes
0answers
11 views

Nnon-trivial derivations on associative algebras

I would like to learn about finite dimensional complex associative algebras with trivial algebra of derivations. For instance, it is easy to see that the algebra $A := span_{\mathbb{C}}\{e_1,\ldots, ...
1
vote
1answer
21 views

A quotient of tensor algebra of $V \otimes k$

Let $k$ be a field and $V$ a $k$-vector space. Denote the inclusion of $k$ in $k \oplus V$ by $m$. A book that I'm consulting (Ramanan's Global Calculus) asks the reader to prove that the quotient of ...
0
votes
0answers
18 views

External doubling of a composition algebra

Let $D$ be a composition algebra over a field $F$. Define on $C = D \oplus D$ a product by, $$(x,y)(u,v)=(xu + \lambda \bar{v}y, vx+y \bar{u}) $$ and a quadratic form $N:C \to F$ by $$N((x,y))= N(x) -...
1
vote
1answer
66 views

Right adjoint of a monad is a comonad

I'm having trouble proving the following statement: If a monad $T$ has a right adjoint $K$, then $K$ is a comonad and the categories of $T$-algebras and $K$-coalgebras are isomorphic. So far I've ...
0
votes
0answers
27 views

Difference between functors $\text{Hom}_B(-,B)$ and $\text{Hom}_{\mathbb C}(-,\mathbb C)$

I am working with a commutative finite-dimensional $\mathbb C$-algebra $B$ and I am supposed to consider and prove a theorem about the hom-functors $$\text{Hom}_B(-,B)\quad\text{and}\quad \text{Hom}_{\...
1
vote
1answer
27 views

Showing the existence of a certain kind of extension of a finitely generated algebra

I wanted to ask if someone had a proof for the following claim: Given and integral domain $A$ and a finitely generated $A$-algebra $B$, show that there exists elements $x_1,...,x_n \in B$ ...
0
votes
1answer
22 views

Why is the supremum of two functions smaller than the individual suprema

Let $X$ denote a topological space, and write $C^b(X)$ for the algebra of bounded, continuous functions on $X$. Why does it then hold that $\left|fg\right|_{X} \leq \left|f\right|_{X} \ \left|g\right|...
0
votes
0answers
50 views

What is the motivation for $R$-algebras?

Dummit & Foote makes the following definition: Let $R$ be a commutative ring with identity. An $\mathbf R$-algebra is a ring $A$ with identity together with a ring homomorphism $f:R\to A$ ...
1
vote
1answer
45 views

Finding a polynomial ring embedded in a fg algebra over complex numbers

I was asked this question and I thought I know how to approach it but I'm completely stuck. The question is as follows: Let $$ A=\mathbb{C}[x_1,x_2,x_3,x_4]/(x_4x_3-x_2x_1, x_1^2x_3-x_4^3x_2) $$ ...
0
votes
0answers
24 views

Operation on ideals of Frobenius algebra

Let $A$ be a Frobenius algebra (i.e. a finite-dimensional, unital, associative algebra equipped with a non-degenerate bilinear form). We define an operation on the ideals of $A$ as follows: $$I \cdot ...
0
votes
2answers
36 views

is $\mathbb{Q}$ finitely generated as a $\mathbb{Z}_{(p)}$-algebra?

I know that $\mathbb{Q}$ is not finitely generated as a $\mathbb{Z}$-algebra (and thus also not finitely generated as a $\mathbb{Z}$-module) how about $\mathbb{Q}$ as $\mathbb{Z}_{(p)}$-algebra? (or ...
0
votes
1answer
34 views

The collection of half open half closed interval with empty set in $\mathbb{R}$ generated a semi-algebra

This question is about Durrett Edition 5 Example 1.1.8, in which he claims that Let $\Omega=\mathbb{R}$, and $\mathcal{S}=\mathcal{S}_{1}$ then $\overline{\mathcal{S}}_{1}=$the empty set put all ...

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