Questions tagged [ra.rings-and-algebras]
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
3,117
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short exact sequence of groups [closed]
We have a short exact sequence as
$$0 \rightarrow \mathbb{Z}_2\rightarrow G \rightarrow \mathbb{Z}_2\rightarrow 0,$$
can we conclude that the group $G$ is isomorphic to $\mathbb{Z}_2 + \mathbb{Z}_2$ ...
3
votes
0
answers
163
views
Coevaluation for linear categories
For a field $k$ and an associative $k$-algebra $R$, the $k$-linear category $R\operatorname{-Mod}$ is self dual inside $\operatorname{DGCat}_k$, with the counit map sending $k$ to $R$ regarded as a ...
- 597
3
votes
2
answers
121
views
Adjunctions and inverse limits of derived categories
Consider a tower $\dots\to A_{2}\to A_{1}$ of rings. This gives rise to a diagram $\mathbb{N}^{\text{op}}\to\text{Cat}_{\infty}$ of $\infty$-categories (confusing $\mathbb{N}^{\text{op}}$ with its ...
- 437
1
vote
0
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62
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Why is equivalent for a sequence to be an $A$-sequence or to be an $A/I$-sequence?
Exercise 24.1 in Matsumura's commutative ring theory:
Let $A$ be a Noetherian ring, and $I$ an ideal of $A$; assume that $I^r=0$, and that $I^i/I^{i+1}$ is a free $A/I$-module for $1\leq i < r$. ...
- 11
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0
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191
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Is the span of all nilpotent ideals also a nilpotent ideal?
Given a non-zero Lie algebra $\mathcal{L}$ over $\mathbb{C}$, we define $\mathcal{L}^2 = \big[\mathcal{L}, \mathcal{L} \big] = \big\{ [x, y]: x, y\in \mathcal{L} \big\}$, and for any $k\in\mathbb{N}$ ...
- 245
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190
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The Game related to 'ring theory' [closed]
I saw the book "Concepts of Modern Mathematics" by Ian Stewart. In this book, I saw the game, 'solitaire game'. the game's rule is like this below contents.
The game of solitaire as you ...
- 15
5
votes
2
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350
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How is the classification of groups extensions by $H^2$ related to Yoneda Ext?
It is well-known that group extensions
$$1\to A \to H \to G \to 1$$
where $A$ is abelian with a $G$-action such that the conjugation action of $G$ on $A$ agree with this fixed action are classified ...
- 2,837
2
votes
0
answers
110
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Generalized wreath products of commutative algebras with Hopf algebras
Fix $k$ a commutative ring (or, if more convenient, assume it’s a field or even an algebraically closed field of characteristic 0, which is the case I’m mainly interested in). Let $A$ be a unital ...
- 1,146
4
votes
0
answers
95
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Lie algebra cohomology of the space of vector fields
For a (closed and oriented) manifold $M$, the first Lie algebra cohomology $H^1(\mathrm{Vect}(M),C^\infty(M))$ of the space of vector fields with coefficients in smooth functions is isomorphic to $H^1(...
- 359
7
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1
answer
1k
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If two Lie algebras are isomorphic, under which conditions will their Lie groups also be isomorphic?
Let $G$ and $G'$ be compact connected Lie groups (which are not necessarily simply connected) with Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$. Suppose that the two Lie algebras are isomorphic, ...
- 89
2
votes
0
answers
178
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On idempotents and local properties of localization of Noether algebras
Let $R$ be a commutative Noetherian local ring. Let $\Lambda$ be a module finite associative $R$-algebra such that $\mathrm{Supp}_R(\Lambda)=\mathrm{Spec}(R)$. Let $\mathfrak p$ be a prime ideal of $...
- 198
8
votes
2
answers
452
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Faithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:
Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
2
votes
0
answers
52
views
Can the Weyl algebra be free over its invariant subalgebra?
Let $k$ be an algebraically closed field of zero characteristic, let $P_n$ denote the polynomial algebra in $n$ indeterminates, and let $G$ be a finite group of linear automorphisms. Then, by ...
- 2,598
5
votes
3
answers
485
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Congruences that aren't "finite from above"
Let $\mathfrak{A}=(A;...)$ be an algebra in the sense of universal algebra. Say that a congruence $\sim$ on $\mathfrak{A}$ is parafinite iff there is an equivalence relation $E\subseteq A^2$ with ...
- 19k
1
vote
1
answer
78
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Formulas for partial composed product
Let $A(x) = \prod\limits_i (x-\lambda_i)$ and $B(x) = \prod\limits_j (x-\mu_j)$. Then, their composed product is defined as
$$
(A*B)(x) = \prod\limits_{i,j} (x-\lambda_i \mu_j).
$$
Generally, we can ...
- 1,131
3
votes
2
answers
238
views
Units of the group algebra of a free group
Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$?
In the case of $n=1$, there are only trivial units: $K[F_1]^\...
- 359
2
votes
1
answer
222
views
Non-example to PBW theorem
I am interested in a (simple) example of an associative algebra $A$ with 1 generated by $x_1, \ldots, x_n$ which is quadratic (i.e. all relations between $x_1, \ldots, x_n$ have degree $\leqslant 2$) ...
- 31
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0
answers
128
views
Elementary equivalence for rings
Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-...
- 2,598
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votes
0
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70
views
Identity for compositum and intersection of fields
Let $k$ be an arbitrary base field and $K, L, M$ some fields over $k$ contained in a fixed overfield $\Omega$.
Question: Are there some "reasonable" assumptions (ie beyond a bunch of really ...
- 5,716
8
votes
1
answer
164
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Do graded-commutative rings satisfy the strong rank condition?
Let $R$ be a ring. Recall that $R$ is said to satisfy the strong rank condition if, whenever $R^m \to R^n$ is a monomorphism of right $R$-modules (with $m,n \in \mathbb N$), we have $m \leq n$.
It is ...
- 59k
2
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0
answers
78
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Automorphism group of the first Weyl field
A related question is this one (Automorphism group of the quantum Weyl field).
Let $A_1$ denote the rank 1 Weyl algebra (over the complex numbers), and $D_1$ its skew field of fractions, called the ...
- 2,598
1
vote
0
answers
81
views
What is a Gelfand-Tsetlin subalgebra?
For context on general Gelfand-Tsetlin theory, see for instance this MO post.
Let's work over $\mathbb{C}$. Fix $n>0$. There is a natural chain of embeddings of the general linear Lie algebras $\...
- 2,598
4
votes
1
answer
268
views
Some folklore about crystaline rings of differential operators
This question is a follow up to my previous question on rings of crystaline differential operators, to which I refer for the adequate definitions.
First, let's consider the case of an algebraically ...
- 2,598
2
votes
0
answers
75
views
An open problem about simple Noetherian rings
The following is a well-known open problem in ring theory (see, for instance, Goodearl, Warfield, 'An introduction to noncommutative Noetherian rings, Appendix, Problem 19)
Question: Let $R$ be a left ...
- 2,598
2
votes
2
answers
374
views
Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)
I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
- 121
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0
answers
238
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Connections in non-commutative geometry
Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...
- 359
5
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0
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180
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Rings where all indecomposable modules are projective or injective
Let $A$ be a semi-perfect noetherian ring.
Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective?
Im also interested in ...
- 25.4k
3
votes
1
answer
185
views
Relation between enveloping algebras and algebras of differential operators
I asked this question on math stack exchange about 3 years ago, but received no answer.
Our base field $\mathsf{k}$ will be algebraically closed of zero characteristic. Let $X$ be an smooth affine ...
- 2,598
2
votes
1
answer
180
views
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 $R$ ...
- 2,598
0
votes
1
answer
89
views
Subalgebras of finite extensions
Suppose that $A\subset B$ is a finite extension of rings. Is it true that every intermediate extension $A\subset C\subset B$ finite over $A$?
- 147
1
vote
0
answers
41
views
Eigenvalues of a subset of matrix semigroup
My apologies for slightly longer post but I wanted to explain lower dimensional cases and their proofs before asking the actual question, which starts after the phrase The general case below.
A two-...
- 111
6
votes
2
answers
1k
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Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
I am posting this question on MO since I haven't received any answers on MSE.
Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...
- 171
4
votes
1
answer
152
views
Nonempty intersection of cosets of finite-index subgroups
$\DeclareMathOperator\lcm{lcm}$This question is crossposted from MSE.
Let $H_1,\dots,H_{n+2}$ be cosets of finite-index subgroups of $\mathbb{Z}^n$ and suppose for all $i=1,\dots,n+2$, $\bigcap_{j\neq ...
- 7,836
3
votes
0
answers
69
views
Vershik's conjecture about generic quadratic algebras
Is it still unknown whether very general (lying in a countable intersection of some Zariski opens in corresponding Grassmannian) quadratic algebras $R$ with $\operatorname{dim} R_2 < \frac{3}{4}(\...
- 4,299
1
vote
2
answers
213
views
Link invariants from Hecke relations of higher order
Alexander theorem says oriented links in $\mathbb{R}^3$ can be
represented by closures of braids. Markov theorem says that
braids related by Markov moves produce isotopic braid closures,
and vice ...
- 4,760
0
votes
0
answers
90
views
Is there a "cohomology theory" for involutive algebras?
I'm aware that there are cohomology theories for algebraic structures related to involutive algebras (or "involution algebras" or "$*$-algebras" if you prefer those terms) like Lie ...
- 4,752
3
votes
1
answer
247
views
The associated graded algebra of a finite dimensional algebra
$\DeclareMathOperator\rad{rad}$Let $A$ be a finite dimensional algebra (we can assume that $A \cong KQ/I$, for a quiver $Q$ and an admissible ideal $I$ if that helps).
Denote by $A_G$ the associated ...
- 25.4k
1
vote
1
answer
207
views
Two exact sequences for $R$-modules: does one imply the other?
Consider the following two properties for an $R$-module $M$:
For every endomorphism $f:M\rightarrow M$, there exist central endomorphisms $g, h\in \operatorname{End}_R(M)$ such that the sequence $M_{...
- 321
4
votes
1
answer
375
views
WZW primary correlations in terms of current algebra?
Given the
$\mathfrak{u}(N)$ algebra
with generators $L^a$ and commutation relations
$ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ ,
the WZW currents of $U(N)_k$
$$ J(z) = \sum_{n \in \mathbb{Z}} J^a_n z^{-n-...
- 525
1
vote
0
answers
102
views
Degrees of trigonometric numbers
For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers.
What is its degree?
That is, what is the minimal degree of ...
- 7,981
2
votes
0
answers
127
views
$p$-adic Banach group algebra
Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
- 1,117
1
vote
0
answers
56
views
A certain subalgebra of $\mathfrak{e}_8$ over a p-adic field
Can $\mathfrak{e}_8(\mathrm{k})$ have a maximal subalgebra isomorphic to $\mathfrak{sl}_1(\mathrm{D})\oplus\mathfrak{g}_2(\mathrm{K})$, where $\mathrm{k}$ is a finite extension of some $\mathbb{Q}_p$, ...
- 2,613
0
votes
0
answers
81
views
Definition of term functions, in universal algebra
According to the definitions in Sankappanavar's universal algebra :
Assume $p$ is a term, then $p(x_1,x_2,...,x_n)$ indicates that the variables occurring in $p$ are among $x_1,...,x_n$. But there is ...
- 11
-1
votes
1
answer
124
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Classification of real Clifford algebras
$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, ...
- 20.5k
2
votes
0
answers
54
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upper bound for the exponential conjugacy growth rate for non-virtually nilpotent polycyclic groups
Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of ...
- 763
5
votes
0
answers
111
views
On the structure of an algebra as a bimodule
$\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ker{Ker}\newcommand{\bi}{\mathrm{bi}}\newcommand{\op}{\mathrm{op}}$Let $K$ be a field (say of characteristic zero), and $...
2
votes
0
answers
138
views
The growth rate of the group $\mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\phi (1)$ corresponds to multiplying every number by $2$
Consider the group $G = \mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\mathbb{Z}[1/2] = \{j/2^m \mid j \in \mathbb{Z}, m\in\mathbb{N} \}$, the dyadic rationals, and for every $n\in \mathbb{Z}$, $...
- 763
5
votes
0
answers
270
views
Serre subcategories of the category of chain complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R$ be a commutative $k$-algebra.
We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...
- 619
2
votes
1
answer
243
views
Gluing data for modules over a ring with idempotents
Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological ...
- 961
4
votes
1
answer
293
views
Algebra with three anti-commutator relations
Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra $V$ generated by $ a,b,c$ and the relations
$$u a^2 = bc + cb$$
$$v b^2 = ac + ca$$
$$w c^2 = ab + ba$$
Is $V$ generated by ...
