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.
367 questions
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Certain conditions on cancellative semigroups
This is extracted from this question following Benjamin Steinberg's suggestion.
For a semigroup $S,$ let $P(S)$ denote the power semigroup of $S,$ which is made up of all non-empty subsets of $S$ ...
- 1,445
3
votes
2
answers
907
views
Finding coefficient of multivariate polynomial
$f(x_1,x_2,\ldots x_n)$ is polynomial with integer coefficients.
$f$ is rather large to be computed explicitly, but an algorithm can
compute it efficiently at integers and complex number and "...
- 25.4k
3
votes
0
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387
views
Software for Combinatorial Algebra sought
I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task:
I have two graded (generally ...
3
votes
1
answer
253
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What is the story behind this Hilbert space in the definition of Hilbert Modules
Here is Deflnition 1.5 of Hilbert module in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert ...
- 2,118
3
votes
1
answer
694
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Reference Request: Basis in terms of ring of symmetric polynomials
As part of the result of solving the problem I am working on, my advisor and I translated the task of finding a basis for $R(T_{sl_{\mathbb{C}}(n)})$ in terms of $R(sl_{\mathbb{C}}(n))$ into the ...
- 799
3
votes
3
answers
357
views
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?
Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...
3
votes
1
answer
328
views
What does it mean for the surreal numbers/partizan games to be "universally embedding"?
In "On numbers and games", Conway writes that the surreal Numbers form a universally embedding totally ordered Field. Later Jacob Lurie proved that (the equivalence classes of) the partizan ...
- 83
3
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2
answers
2k
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Extension problem
As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...
- 2,857
3
votes
1
answer
158
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Local Frobenius algebras and their opposite algebras
Assume all algebras are finite dimensional quiver algebras over a field (no restriction of generaltiy if the field is algebraically closed).
Let A be a local Frobenius algebra.
Is A isomorphic to its ...
- 26.5k
3
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0
answers
67
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Atomicity and BF-ness in monoids of integer points of a polyhedral cone of $\mathbb R^n$
Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is ...
- 10.5k
3
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1
answer
564
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Is there a classification of the $p$-adic normed division algebras?
A normed division algebra over $\mathbb{R}$ is a pair $(A,\lVert{-}\rVert)$ with
$A$ an $\mathbb{R}$-algebra with a unit $1_A$;
$\lVert{-}\rVert\colon A\to\mathbb{R}_{\geq0}$ a norm on $A$;
such ...
- 11.8k
3
votes
1
answer
186
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If a PID has no nonzero divisible elements, then is the same true of its finitely-generated modules?
EDIT: The question was originally about general Noetherian rings instead of PID's. Thanks to YCor for pointing out how wrong this was in the comments below (1 2 3).
Question 1: Let $R$ be a PID. ...
- 63.9k
3
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0
answers
93
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Is a specific endomorphism of $A_1$ an automorphism?
Let $k$ be a field of characteristic zero, and let $A_1(k)$ be the first Weyl algebra, namely, the associative non-commutative $k$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$.
...
- 2,837
3
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1
answer
336
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Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$.
The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$
has the bracket
$$[xt^r, yt^...
- 138
3
votes
1
answer
343
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How to define a generating subset for algebra in a category?
As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...
3
votes
1
answer
224
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Rings or algebras with many nilpotent elements and efficient computation
Crossposted from quantum.SE
where comment appears to suggest that solving modulo 2 might
be possible.
Searching the web for '"quantum computer" nilpotent'
returns many results, so maybe the ...
- 25.4k
3
votes
1
answer
141
views
The "semi-symmetric" algebra of a vector space
If $V$ is a vector space over a field $K$, then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\...
3
votes
1
answer
472
views
Tips for how I can proceed with my Lie theoretical problem?
$\DeclareMathOperator\SL{SL}$I am looking at a map from a Lie group into a Lie algebra $\phi$:
$$\phi: \SL(n)\rightarrow \mathfrak{sl}_n$$
$$ P \rightarrow U_1^\dagger P U_1 + U_2^\dagger P U_2.$$
$P$ ...
3
votes
2
answers
243
views
Minimal fields of isomorphism for varieties
Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
- 11.3k
3
votes
3
answers
435
views
Is there a noncommutative simple fusion ring?
A fusion ring $\mathcal{F}$ is given by a finite set $B = \{b_1,b_2, \dots, b_r \}$ such that $b_i b_j = \sum_k n_{i,j}^k b_k$ with $n_{i,j}^k \in \mathbb{Z}_{\ge 0}$, satisfying axioms slightly ...
3
votes
4
answers
2k
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Reflexive modules over a 2-dimensional regular local ring
This question comes from reading Washington's proof of Iwasawa's theorem, and wanting to learn the commutative algebra version of the classification of finitely-generated $\Lambda$-modules. I went to ...
- 387
2
votes
1
answer
643
views
Linear dependency of real numbers with integer coefficients adding up to zero [closed]
Let $x = (x_1, \dots, x_n)$ be a vector of real number. I was asking myself if there was an efficient way of telling whether there exists a non-zero vector of integers $z \in \mathbb Z$ such that both
...
user45183
2
votes
1
answer
514
views
When does $R [x]/I $ has infinitely many idempotents?
Let $R$ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotents and an unextended ideal $I$ in $R[x]$ such that $R[x]/I$ has ...
- 51
2
votes
0
answers
208
views
A functor on the category of rings, algebras or compact Hausdorff topological space
Assume that $R$ is a unital ring or a complex or real (Banach or $C^{*}$) algebra.
We define a relation $M$ on $R$ as follows: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...
- 356
2
votes
1
answer
151
views
Automorphism of algebras with certain initial conditions on given idempotents
The First question
Let $A$ be a Banach or a $C^*$ algebra. Assume that $e,f$ are two idempotents or prjections in $A$ which satisfy $ef=fe=0$. Assume that there are two automorphisms $\phi, \psi: A \...
- 356
2
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1
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122
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If $H$ is essentially equimorphic to $K$, then is $H$ atomic only if so is $K$?
I will first state my question, and then give all the relevant definitions.
Q. Let $H$ and $K$ be monoids, and assume $H$ is essentially equimorphic to $K$. Is it true that $H$ is atomic only if so ...
- 10.5k
2
votes
2
answers
234
views
Chain of ideals in a BA
Suppose $\mathfrak{A}$ is a Boolean algebra and $\mathfrak{J}$ is chain of ideals in $\mathfrak{A}$ ordered by inclusion such that none of its elements is countably generated. Clearly, the union $\...
- 31
2
votes
1
answer
224
views
Finite rank ring
Let given ring $R$ of finite rank. Is it true that for all primes $p$ large enough modules $Der_{\mathbb{Z}}(R/pR) = \{0\}$?
For every ring we define $Der_{\mathbb{Z}}(R)$ as set of linear operators $...
- 21
2
votes
2
answers
673
views
polarization/linearization as in jordan forms
I am new to this branch of math, so bear with me.
This question started when reading Kevin McCrimmon's "A Taste of Jordan Algebras"
It talks about polarization and gives a general ...
- 23
2
votes
1
answer
192
views
Origins of a theorem on an atomic factorizations in domains and cancellative monoids satisfying the ACCPL and the ACCPR
Let $H$ be a (commutative or non-commutative) monoid. We say that $H$ satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left ...
- 10.5k
2
votes
1
answer
290
views
Endomorphism rings of infinitely generated free modules generated by idempotents?
Let $M$ be a free right $R$-module. When $M_R\cong R_R^n$ with $n\in \mathbb{Z}_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M_R)$ is isomorphic to $\mathbb{M}_n(R)$. We also know ...
- 23
2
votes
1
answer
445
views
Algebra of endomorphisms of f.g. modules as subquotients of matrix algebras
Let $A$ be a $C$-algebra, where $C$ is a commutative ring with $1$, and $M$ be a finitely generated left $A$-module.
Question: Is it true that we can always find a positive integer $n$, a $C$-...
- 279
2
votes
0
answers
95
views
Socle of a quotient of the ring of differential operators of a polynomial ring
I have been reading the following paper:
https://www.sciencedirect.com/science/article/pii/S002240491000263X
Proposition 2.4(ii) shows that if $\mathfrak D$ is a ring of $k$-linear differential ...
- 81
2
votes
2
answers
757
views
Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
- 33
2
votes
1
answer
138
views
Associated graded algebras and symmetric Frobenius algebras
Let $A$ be a filtered algebra and let $G$ be its associated graded algebra. As discussed in this question, if $G$ is Frobenius, then $A$ is also Frobenius.
If $G$ is a symmetric Frobenius algebra, ...
- 157
2
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1
answer
1k
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Determinant of integer lattice basis of `$L=\{(x_1,\ldots,x_n): a_1x_1+\cdots+a_nx_n=0\}$`
Question: Suppose $\{v_1,\ldots,v_{n-1}\}$ is
an integer basis for the lattice
$$L=\{(x_1,\ldots,x_n)\in\mathbb{Z}^n:
> a_1x_1+\cdots+a_nx_n=0\},$$ where the
$a_i$ are fixed nonzero integers. ...
- 430
2
votes
0
answers
120
views
A Boolean quotient ring of a prime ring
I am searching for a unital prime ring $R$ such that its right socle $Soc(R_R)$ is nonzero and proper, and such that $R/Soc(R_R)$ is a Boolean ring (i.e., all its elements are idempotent).
Thanks for ...
- 355
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0
answers
98
views
Do $r(a) \leq^\oplus R$ and $r(a) = r(a^2)$ imply $r(a) = eR$ and $aR \subseteq (1-e)R$ for some idempotent $e$?
Let $R$ be a (commutative or non-commutative, associative) ring with unity, and let $a$ be an element of $R$ such that $r(a) = r(a^2)$, where $r(\cdot)$ denotes a right annihilator. It follows that $r(...
- 10.5k
1
vote
1
answer
120
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"Coloring" the ideals of a ring
Let $R$ be a ring with more than 1 element, and let $A$ be a non-empty set. We call a map $c:R\to A$ an ideal coloring if for every nonempty ideal $I$ with $I\neq\{0\}$ the restriction $c|_I$ is not ...
- 48.9k
1
vote
1
answer
234
views
Zeroes of elementary polynomials without involving closed-form solutions
Consider the following two polynomials, where $n$ is an integer:
$$
p_n(x) = x^3-\frac1nx-\frac2n, \\
q_n(x) = x^2-\frac2n.
$$
For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
- 21
1
vote
1
answer
263
views
Relationship between units of a ring and primitive characters of the ring under addition
Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian.
My question is the following problem:
Given the dual group $\...
1
vote
0
answers
73
views
Relationship between units and primitive characters 2
This is a follow up to this question.
Let $(R,+,\cdot)$ be a finite ring.
Definition Given the dual group $\widehat{R}$ of $(R,+)$, a character $\chi\in\widehat{R}$ is said to be primitive with ...
1
vote
1
answer
96
views
If $H$ is commutative and unit-cancellative, then so is the monoid of non-empty ideals of $H$
Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take
$$XY := \{xy: x \in X,\, y \in Y\}.$$
We call a set $I \subseteq H$ an ideal of $H$ ...
- 10.5k
1
vote
0
answers
294
views
Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
1
vote
1
answer
157
views
Non-degeneracy of comultiplication (multiplier Hopf algebras)
Consider the following fragment from the paper "Multiplier Hopf-algebras" by Van Daele.
Can someone explain how the coassociativity in definition 2.2 (ii) and the requirement $(\Delta \...
user167952
1
vote
0
answers
74
views
Construction, similar to Chow's EL-numbers? Is it valid? What are the properties?
The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out.
Instead of $\exp(x)$ and $\ln(x)$ functions as the building ...
- 10.1k
1
vote
0
answers
234
views
Separability of a simple ring extension
Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
- 2,837
1
vote
1
answer
285
views
a problem about ideals of polynomial rings
Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions
$$
f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i.
$$
Let $(...
- 1,990
1
vote
1
answer
195
views
Strongly graded algebras with no zero divisors
Let $A = \bigoplus_{i \in \mathbb{Z}} A_i$ be a strongly graded unital algebra over $\mathbb{C}$, with no zero divisors. Is it always true that
$$
m: A_i \otimes_{A_0} A_j \to A_{i+j}
$$
is an ...
1
vote
0
answers
164
views
When a finite codimensional subalgebra contains a finite codimension ideal?
What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property:
Every finite codimensional subalgebra $B$ of $A$ ...
- 356