All Questions
Tagged with or ra.rings-and-algebras ra.rings-and-algebras
3,500 questions
25
votes
2
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What is the name of this relative semidirect product of groups?
We have two well known definitions of the semidirect product $N \rtimes H$ of groups:
(Internal semidirect product) We write $G = N \rtimes H$ if $N$ is a normal subgroup of $G$, $H$ is another ...
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 28.7k
2
votes
2
answers
416
views
Tensor product over $\mathbb{Z}$ and p-adic integer ring $\mathbb{Z}_p$
Thanks for your reading. Suppose we have two $\mathbb{Z}_p$-modules $A,B$. Do we always have $A \otimes_{\mathbb{Z}} B \simeq A \otimes_{\mathbb{Z}_p} B$, as abelian groups or $\mathbb{Z}_p$-modules? ...
CommunityBot
- 1
5
votes
0
answers
140
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Gelfand-Kirillov dimension and tensor products
$\DeclareMathOperator\GK{GK}$Let $k$ be the base field.
The Gelfand-Kirillov dimension was introduced by Gelfand and Kirillov in their seminal paper on the Gelfand-Kirillov conjecture.
A very famous ...
- 3,318
5
votes
2
answers
199
views
Determining the multiplication via addition and some unary operation
It is known that the addition operation in a skew-field $F$ (more generally, in a quasifield) is uniquely determined by the multiplication operation and the unary involutive operation $1_{-}:F\to F$, ...
- 14.6k
23
votes
2
answers
2k
views
Structures of the space of neural networks
A neural network can be considered as a function
$$\mathbf{R}^m\to\mathbf{R}^n\quad
\text{by}\quad x\mapsto w_N\sigma(h_{N-1}+w_{N-1}\sigma(\dotso h_2+w_2\sigma(h_1+w_1 x)\dotso)),$$
where the $w_i$ ...
- 3,599
3
votes
1
answer
188
views
Is there a uniform family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ such that $f_p(x)\in \mathbb{Z}[x]$ is irreducible and irreducible mod $p$?
Let $p\in\mathbb{Z}$ be a positive prime number.
Is there a "uniform" family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ of degree two such that $f_p(x)\in \mathbb{Z}[x]$ is both irreducible ...
- 704
0
votes
0
answers
61
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The theory of Groebner bases in Jordan case
There are many papers regarding spreading the theory of Groebner-Shirshov bases from Lie algebras to other nonassociative algebras. Also, it has been studied for associative algebras with operators ...
- 63.9k
2
votes
0
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187
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How to construct an explicit isomorphism of the split Quaternion Algebra $(a,b)_F$ over the field $F$ to $\mathrm{Mat}_2(F)$
$\DeclareMathOperator\Mat{Mat}$How to construct an explicit isomorphism of the split quaternion algebra $(a,b)_F$ over the field $F$ to $\Mat_2(F)$?
As it is known that the algebra of quaternions is ...
CommunityBot
- 1
2
votes
0
answers
33
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R be a semiprime ring, not right singular, and with finitely uniform dimension to the right
if R is a semiprime ring, not right singular, and with finitely uniform dimension to the right. how can I Show that if I is a non-zero right ideal of R, then there exists an element y ∈ I such that
I ∩...
- 21
3
votes
0
answers
106
views
Formulas for the line joining two points in the projective plane over a division algebra
Let $K$ be a[n associative] division algebra (= skew field). By the “projective plane” $\mathbb{P}^2(K)$ over $K$ I mean, as usual, the set of triples $(x,y,z)$ of elements of $K$, not all zero, up ...
- 32.4k
0
votes
0
answers
81
views
Can every $\ast$-algebra be represented in this space of matrices?
Let $k$ be a field with characteristic $0$. For every set $X$, let $\mathcal{B}(X)$ be the set of (possibly infinite) matrices $T = (T_{x,y})_{x,y \in X}$ with coefficients in $k$ such that in each ...
- 1,485
3
votes
0
answers
100
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Pairing on a Koszul dual pair
Let $A$ be a graded quadratic algebra over a field $k$, and suppose that it admits the Koszul dual $A^!$. I want to know if there is a natural pairing $A\otimes A^!\to k$ or something similar to this. ...
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9
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2
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471
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Can any properties of a ring other than being a field be captured by the geometry of its 2-dimensional free module?
Can any properties of a ring other than being a field be captured by the geometry of its 2-dimensional free module?
Background:
In his wonderful, wonderful book Geometric Algebra, Emil Artin describes ...
- 4,713
1
vote
0
answers
37
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Bounding the length of an R-module of matrices
Loosely related to this: Bounding the length in a module of evaluated skew polynomials
Let $C$ be an $\mathbb{F}_q$-vector subspace of $m \times n$ matrices over $\mathbb{F}_q$. Assume WLOG that $m \...
- 63.9k
31
votes
3
answers
3k
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Infinite-dimensional normed division algebras
Let's say a normed division algebra is a real vector space $A$ equipped with a bilinear product, an element $1$ such that $1a = a = a1$, and a norm obeying $|ab| = |a| |b|$.
There are only four ...
- 1,446
1
vote
1
answer
363
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Proj construction and nilpotent homogenous elements in graded ring
Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined ...
- 5,998
2
votes
1
answer
124
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Proof of dynamic programming calculation of Levenshtein distance
Let s1 and s2 are 2 arbitrary strings with lengths l1 and ...
- 981
0
votes
0
answers
85
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A naive looking question about Gelfand-Kirillov dimension
Let $A$ and $B$ two affine algebras, $A$ a subalgebra of $B$. If we have a left $A$-module $M$ we can extend the scalars: $B \otimes_A M$. I will denote the resulting $B$-module by $N$
How are $\...
- 35.2k
1
vote
0
answers
162
views
Integral points on "complex exponential surface" in $\mathbb{C}^3$
I encountered the following object in $\mathbb{C}^3$ defined for $m\in\mathbb{N}$ by
$$A_m:=\lbrace (z_1,z_2,z_3)\in\mathbb{C}^3|(2^{2z_3}m-1)2^{2z_1+z_2+1}+3^{z_2-1}(2^{2z_1}-2^2-3^{z_3+1}m)=0\rbrace$...
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-1
votes
1
answer
209
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Every abelian group can be embedded into a ring [closed]
Let $(G,0,+)$ be an abelian group. Does there always exist a ring with unity $(R,0,1,+,\cdot)$ and an injective homomorphism of groups $ \psi:(G,0,+)\rightarrow (R,0,+)$?
Is this hard to prove, or are ...
- 3,054
1
vote
0
answers
118
views
Cyclic homology with coefficients in a bimodule
I've recently been trying to understand Hochschild and cyclic co/homology better, and I've noticed that while it's common to define the Hochschild homology $\mathrm{HH}_{\bullet}(A;M)$ of an $R$-...
- 11.8k
2
votes
2
answers
345
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Are there atomistic ortholattices which are not modular?
Let $L$ be an atomic ortholattice. We say that two elements $a$ and $b$ of $L$ are orthogonal if $a\leq b^\perp$. If $L$ is orthomodular then every element of $L$ can be written as a join of pairwise ...
- 4,219
6
votes
1
answer
273
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Let $f$, $g$ be two complex polynomials satisfying $f(x-1,y)g(x,y)=f(x,y)g(x-1,y-1)$, what we can say about $f$ and $g$?
Let $f$, $g$ be two non-zero polynomials in $\Bbb C[x,y]$ satisfying the identity
$$
f(x-1,y)g(x,y)=f(x,y)g(x-1,y-1).
$$
What we can say about $f$ and $g$?
In particular, if $f$, $g$ are two non-zero ...
- 105k
1
vote
0
answers
77
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$M^2=0$ defines a Koszul algebra ? What if $M$ is Manin's endomorphism's of Koszul $A$ ? (Here $M^2=\sum_k M_{ik}M_{kj}$ - resembles $d^2=0$).)
Consider a matrix $M$ which elements $M_{ij}$ are generators of some algebra $K$,
impose new relations: $M^2=0$ and get a new algebra $K_{2}$.
Question 1: Is it true that $K_2$ is Koszul algebra when ...
- 24.8k
1
vote
0
answers
60
views
Bounding the length in a module of evaluated skew polynomials
Let $R$ be a finite principal ideal ring, $S$ a Galois extension of $R$ of degree $m$ (so in particular $S$ is a free $R$-module of rank $m$, and we have an $R$-module isomorphism $S^n \cong \...
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14
votes
3
answers
1k
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About enveloping algebras of direct sums
This question is imported from MSE. It is linked to this one in the case of semi-direct products.
My question Let us consider a Lie $R$-algebra ($R$ is a commutative ring) written as a (module) ...
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11
votes
0
answers
436
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
1
vote
0
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189
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The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 30.1k
3
votes
0
answers
40
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Filtering a pre-Koszul algebra to get a homogeneous associated graded algebra
In Priddy's paper "Koszul resolutions", on p. 42 he defines an algebra $A$ to be pre-Koszul if it can be written as a quotient of a free algebra $F = F\langle x_i \rangle$ with generators $\{...
- 4,254
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votes
0
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34
views
Support of a function acting on an algebra?
Quick: for a measurable function $f$ its support on Euclidean space is clearly just the subset where $f$ does not vanish.
Now, let’s have $f$ acting on an finite Lie algebra, f.e. $\mathfrak{gl}$ as $...
0
votes
0
answers
71
views
When is a submodule trivial?
I am a beginner concerning module theory, but I need it for my PhD. Sorry for obvious questions therefore.
Given a left $C(G)$-module $(V, \tilde{\rho})$ where $C(G)$ denotes the group algebra over a ...
5
votes
2
answers
397
views
Ring with vanishing $K_0$
Suppose we have a ring $R$ such that the Grothendieck group $K_{0}(R)=0$.
Question 1: Does it follow that there exists two positive natural numbers $n\neq m$ such that
$R^{m}$ is isomorphic to $ R^{n}$...
- 7,893
9
votes
2
answers
977
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Topological problems solved by lattice duality
It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...
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0
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106
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Is there a counterpart to "a group acting on a space" to "a ring acting on a space", especially if the space is a (Lie) algebra?
I have read something that comes close to this is a module, but as I have understood, a module requires my space to be an abelian group, which would not be the case for a Lie algebra.
So, if I have ...
0
votes
0
answers
46
views
submodules in a direct sum of semisimple modules without common simple factors
Let $A$ be an associative (unital) algebra. Let $M_1,\cdots, M_r$ be pairwise non-isomorphic simple $A$-modules and let $V=\bigoplus^r_{i=1}V_i$, where
$$
V_i=M_{i,1}\oplus \cdots\oplus M_{i,n_i}\...
- 620
3
votes
1
answer
279
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Wedderburn–Artin like theorem for infinite dimensional Lie algebras?
The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings.
Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ ...
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0
votes
0
answers
64
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Proof of a folkloric result about PI-algebras [duplicate]
I am not not an specialist in PI-algebras, but I can say I have a rather good understanding on the subject.
It is, of course, interesting to discover if an algebra $A$ is a PI-algebra. But it is also ...
- 3,318
9
votes
0
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192
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Where is it shown that a countable self-injective ring is semilocal?
In Lawrence, John. "A countable self-injective ring is quasi-Frobenius." Proceedings of the American Mathematical Society (1977): 217-220. the first line is this:
...
- 1,507
0
votes
0
answers
124
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Do unitary adjoint representations on $\mathfrak{sl}$ form a ring?
I am not too deep into abstract algebra, but I need it badly for my PhD. Therefore, I would be happy for some help here!
I try to give some sense into:
Let $\mathrm{Ad}_{g*}M= U^\dagger M U$ be a ...
2
votes
2
answers
1k
views
Rank of a linear combination of quadratic forms
Suppose we have a set of quadratic forms $Q_i (x_1, \dots, x_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that ...
- 50.6k
0
votes
0
answers
33
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determinantal ideal of sum of Galois conjugate matrices
Given $n$ matrices $A_i \in \mathbb{Z}^{m\times m}$. I am interested in the ideal $I_d(A)$ generated by the $d\times d$-minors of $A = \sum_{i=1}^n x_iA_i \in \mathbb{Z}[x_1, \dots , x_n]$.
The matrix ...
- 61
3
votes
1
answer
426
views
Is Malcev completion an embedding?
The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by
$$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$
the group-like part of the completed (by the augmentation ...
- 8,524
0
votes
0
answers
103
views
Matrix of the minimal projective presentation of a $\tau$-rigid module
Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the ...
- 839
0
votes
1
answer
54
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Orthogonality in Hilbert algebras and congruence
Consider a finite-dimensional Hilbert space $V$ (say, over $\mathbb{C}$) and a finite-dimensional Hilbert algebra $A$ (i.e., Hilbert space with a compatible associative unitary algebra structure). ...
- 42.8k
12
votes
0
answers
224
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Do compact inverse-property loops (or just compact Moufang loops) have invariant uniformities and bi-invariant Haar measure?
So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left and right invariant? (And we can restrict to ...
- 2,585
4
votes
0
answers
118
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Adjoining new factors for primes in UFDs
It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...
- 18.7k
2
votes
1
answer
158
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How to decompose a given polynomial by ideal generators
Given a finite set of polynomials $f_1, f_2,..., f_n$ of variables $x_1,...,x_m$, generating the ideal $I$, suppose that we have one more polynomial $g\in I$.
What is the algorythm for decomposing $g$ ...
- 34.3k
6
votes
3
answers
434
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What is known about finite dimensional modules over the nilCoxeter algebra?
Recall that the nilCoxeter algebra $\mathcal{N}_W$ for a Coxeter group $W$ is given by the $\mathbf{k}$-basis $x_w$ for each $w\in W$ and multiplication $x_ux_v=x_{uv}$ if $\ell(uv)=\ell(u)+\ell(v)$ ...
- 26.5k
1
vote
0
answers
148
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Vanishing (infinite) tensor products
Since the advent of free probabilities and QFT, infinite tensor products of $R$-associative algebras with units has become more familiar to the working mathematician.
Starting from the (permuting) ...
- 3,738