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.

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Isn't every algebraic operad equipped with a trivial weight?

In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem): Let $P$ be a connected weight graded differential ...
groupoid's user avatar
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1 vote
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34 views

An infinitely generated Lie algebra, its finitely generated envelope

If a Lie algebra $\mathfrak g$ is finitely generated, its enveloing algebra $U\mathfrak g$ is finitely generated as an associative algebra. In fact, taking the enveloping algebra of the surjection $\...
Qwert Otto's user avatar
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46 views

On a lemma of projective dimension

Let $R$ be a finite-dimensional algebra, and $A=R\oplus A_1\oplus A_2\oplus \dotsb$ be an $\mathbb{N}$-graded algebra which is locally finite (i.e. all $A_i$'s are of finite dimension). Let $\text{...
Noto_Ootori's user avatar
1 vote
2 answers
105 views

Property for bounding matrix exponential

Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices $X$, $Y$ it holds that $\|e^{X+Y}-e^{X}\| \leq \|Y\|e^{\|X\|} e^{\|Y\|}$ where $\|\cdot\|$ ...
KatsanikJr's user avatar
0 votes
0 answers
92 views

A question about the existence of rational functions

I am reading a paper Representations of shifted quantum affine algebras. I have a question about the existence of a rational function about the remark $4.4$ I'll briefly describe the problem. We let $...
fusheng's user avatar
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11 votes
2 answers
502 views

Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?

Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates? Playing around with the case $n = 2$, I’m pretty sure ...
Tim Campion's user avatar
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7 votes
1 answer
271 views

Zero divisors in an algebra with two generators

Let $k$ be a field, and $R = k\langle x,y \mid x^2 = 0\rangle$. The generators $x$ and $y$ are not supposed to commute with each other. Is the only case where nonzero elements $a, b \in R$ satisfy $...
Ralle's user avatar
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2 votes
1 answer
163 views

Finite étale cover of factorial ring

Let $A$ be a regular factorial ring. Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
prochet's user avatar
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2 votes
0 answers
86 views

On the irreducible submodules of adjoint representations $\text{ad}^{0}$

Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
stupid boy's user avatar
11 votes
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434 views

Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry

In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry. For graphs this had been an open ...
David Roberson's user avatar
1 vote
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22 views

Inner product of signatures of piecewise linear paths

It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$...
Gaspar's user avatar
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6 votes
1 answer
369 views

Tame-Wild dichotomy; why can't tame algebras be wild?

I would like to understand the Tame-Wild dichotomy, and in particular why an algebra cannot be tame and (semi-)wild at the same time. I've looked in the papers by Drozd and Crawley-Boevey [D80, CB88]. ...
Jacob FG's user avatar
  • 477
2 votes
2 answers
66 views

Reference request for a subfamily of regular graphs

[Repost of same question math stack exchange which got no answers] I'm looking for literature on the following family of graphs: Call a regular graph $G=(V,E)$ (of regularity degree $d$) nice if there ...
jojo's user avatar
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0 answers
54 views

Names for product-like algebras involving a "duo of directed pseudoforests"

I am looking for the names (and/or for any information regarding) two algebras, one "free" and one "restricted" by an equivalence class. In both cases, there is an (infix) binary ...
user1661473's user avatar
2 votes
0 answers
142 views

Resolutions of semi free (or almost free) commutative dg algebras with finitely generated cohomology

Let $A^{\bullet}:=\{ \cdots \rightarrow A^i \overset{d^i}{\rightarrow} A^{i+1} \rightarrow \cdots \rightarrow A^{-1} \rightarrow A^0 \rightarrow 0 \rightarrow \cdots \}$ be a non-positively graded ...
Walterfield's user avatar
4 votes
0 answers
377 views

Problem 1.8 from Kirby's list

Context I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
Amanuel Jissa's user avatar
17 votes
0 answers
1k views

Relations in a certain Lie algebra

Let ${\mathfrak g}$ be the (real) Lie algebra generated by infinitely many generators $D_i, E_i$ with $i=1,2,3,\dots$ subject to the following relations for any natural numbers $i,j$: \begin{gather*} [...
Terry Tao's user avatar
  • 108k
9 votes
1 answer
193 views

Matrix ring isomorphisms of different sizes

Do there exist (unital, associative, noncommutative) rings $R$ and $S$, where $\mathbb{M}_2(R)\cong \mathbb{M}_3(S)$, but these matrix rings are not isomorphic to $\mathbb{M}_6(T)$ for any ring $T$?
Pace Nielsen's user avatar
3 votes
0 answers
121 views

Composition of Frobenius $n$-homomorphisms, characteristic-free?

This question is, as so often, a crossbreed of curiosity and laziness. The former has led me to an interesting, but somewhat unsatisfactory paper by Khudaverdian and Voronov (arXiv:2002.02395v2) and, ...
darij grinberg's user avatar
2 votes
1 answer
64 views

Generating sets for a module and scalar extension

Let $k$ be an algebraically closed field and $K/k$ a (transcendental) field extension. Let $A$ be a finite dimensional $k$-algebra, and $M$ an $A$-module. Suppose that the $K \otimes_k A$-module $K \...
bm3253's user avatar
  • 23
0 votes
0 answers
88 views

Isomorphism in division algebras

Let $D$ be a division algebra with center $F$ and $D'$ a division algebra with center $K$, where $K$ is a Galois field extension over $F$. Let $\phi: D \otimes K \rightarrow D'$ be $K$ algebra ...
Jyoti's user avatar
  • 11
0 votes
0 answers
63 views

Hensel lifting of roots of a biquadratic polynomial

Let $5$ divide $p-1$. Therefore, we have $$1+x+x^2+x^3+x^4=(x-\alpha)(x-{\alpha}^2)(x-\alpha^3)(x-\alpha^4)=f_1f_2f_3f_4$$ over $F_p,$ where $\alpha$ is an element of order $5$ in ${F_p}^\times.$ We ...
HIMANSHU's user avatar
  • 381
3 votes
0 answers
100 views

Lie algebra cohomology of formal non-commutative vector fields

Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
Qwert Otto's user avatar
2 votes
1 answer
136 views

The presentations of finite complete local rings

Suppose that $R$ is a commutative ring such that there is a surjection $ \pi:\mathbf{Z}_p[[T_1,\cdots,T_n]]\to R$ of rings where $\mathbf{Z}_p[[T_1,\cdots,T_n]]$ is the ring of formal power series ...
stupid boy's user avatar
2 votes
0 answers
85 views

A recursive description of the smallest divisor-closed subsemigroup containing a set

Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid). ...
Salvo Tringali's user avatar
10 votes
2 answers
483 views

Isomorphic finite fields of a skew field

Let $D$ be a skew field and $F$ and $E$ be isomorphic finite subfields of $D$, is it true that $F=E$?
Alborz Azarang's user avatar
4 votes
1 answer
185 views

Does hereditary and connected imply that the underlying ring $k$ of a $k$-algebra is a field?

All rings are assumed to be associative and have a 1. Let $k$ be a commutative artininan ring and $R$ a finitely generated $k$-algebra. Is it true that if $R$ is connected and hereditary, then $k$ is ...
kevkev1695's user avatar
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3 votes
1 answer
150 views

Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational

Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $f \colon H \to K$ is a (...
Salvo Tringali's user avatar
1 vote
1 answer
165 views

Matrices over a finite field: matrices for which some unipotent $U$ satisfies Trace$(ZU)=0$ for all $Z$ in the commutant

Let $p$ be an odd prime number, let $A\in M_p(\mathbb{F}_p)$ be a $p$-by-$p$ matrix with coefficients in $\mathbb{F}_p$, let $C(A)$ be the commutant of $A$, and let $N\in M_p(\mathbb{F}_p)$ be a ...
loup blanc's user avatar
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5 votes
0 answers
177 views

From group ring to ring ring?

For a group $G$, the set $\mathbb{Z}[G]$ of all formal $\mathbb{Z}$ linear combinations is a ring with unit. Now the set $\mathbb{Z}[\mathbb{Z}[G]]$ gets the structure of a ring from the addition in $\...
Nandor's user avatar
  • 289
3 votes
1 answer
192 views

Tangent space of a GIT quotient of $GL_{N}$

Let $G:=\operatorname{GL}_{N}$ act on its Lie algebra $\mathfrak{g}:=\mathfrak{gl}_{N}$ by conjugation. Then it acts naturally on the associated ring $\mathcal{O}(\mathfrak{g})$ of (algebraic or ...
Shaul Zemel's user avatar
1 vote
1 answer
215 views

Wedderburn theorem for finite-dimensional algebras over the complex numbers

I'm trying to understand how to apply the Wedderburn theorem in the context of unitary algebras over $\mathbb{C}$ that are finite-dimensional and semisimple. Let $\mathcal{A}$ be a $\mathbb{C}$-...
Henrique Assumpção's user avatar
1 vote
0 answers
106 views

Is there a "natural" interpretation of the power function for octonions and for sedenions?

This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$, it is impossible to define $\log(x)$ ...
Dieter Kadelka's user avatar
9 votes
1 answer
216 views

Formal smoothness of path algebras and connections

Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if $$ \Omega^1_kA = \operatorname{Ker}(\...
Qwert Otto's user avatar
2 votes
2 answers
161 views

Minimal ideals and subalgebras of semisimple algebras

I'm considering an algebra to be a ring which is also a vector space over some field $F$, and the algebra $A$ is said to be semisimple if it is semisimple as a ring, i.e., $A$ can be written as a ...
Henrique Assumpção's user avatar
5 votes
1 answer
236 views

Pairwise orthogonality for partitions of unity in a *-algebra

Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
JP McCarthy's user avatar
3 votes
1 answer
139 views

Kernels and cokernels in a quotient of an abelian category

I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
Ji Woong Park's user avatar
4 votes
0 answers
291 views

Are there infinitely many simple integral fusion rings of rank $4$?

$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i)...
Sebastien Palcoux's user avatar
-1 votes
1 answer
208 views

Can we classify all commutative unital algebras over the reals that are closed under $\sqrt{}$?

Can we classify all finite dimensional commutative (but not necessarily associative) unital algebras over the reals in which every element is a square?
mick's user avatar
  • 721
1 vote
0 answers
56 views

Matroid for Laurent series

I am trying to find a matroid for profinite rings which are the inverse limit of their finite quotients, and whose linearly independent elements are of the form $L((t_1,\dots,t_n))$. To set this up, ...
Ronald J. Zallman's user avatar
1 vote
0 answers
45 views

generating set of polynomial ring

I am considering the polynomials $P=P[x_1,x_2,\ldots,x_n]$ with coefficients in a ring $R$. Consider a subset $S=\{p_1,p_2,\ldots,p_k\}$ of $P$. There is a map $f\colon P[x_1,x_2,\ldots,x_k] \to P$ ...
David Hillman's user avatar
1 vote
0 answers
62 views

Groups with prescribed Ulm invariants

In Kaplansky's book infinite abelian groups he provides (through some exercises) a complete classification of $p^{\infty}$-torsion countable abelian groups in terms of Ulm invariants. In other words ...
Richard's user avatar
  • 11
0 votes
1 answer
60 views

Left quasi-inverse elements: motivation

An element $a$ in a ring $R$ is a left quasi-inverse if there exists $b\in R$ such that $a+b=ba.$ What is the motivation behind this definition?
spohreis's user avatar
  • 123
3 votes
0 answers
160 views

Amalgamation of commutative subrings

Let $A$ and $B$ be commutative subrings of a non-commutative ring $X$. Is there always a commutative ring $Y$ containing $A$ and $B$ preserving their intersection? This is equivalent to ask if in the ...
user520947's user avatar
1 vote
1 answer
227 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 ...
chrisv's user avatar
  • 21
0 votes
0 answers
40 views

Nonassociativity in Cayley-Algebras

Let $(E,s)$ be a Cayley algebra over a unital commutative ring $A$ with unit element $e$ and $s$ an antiautomorphism (i.e. $s(uv) = s(v)s(u)$, $u,v \in E$) of $E$ such that $u + s(u) \in Ae$ and $N(u) ...
Dieter Kadelka's user avatar
3 votes
0 answers
105 views

On the conditions for Artin-Schelter Gorenstein algebras

Let $ k $ be a field and $ A $ a connected graded $ k $-algebra ($ A $ is associative, but not assumed to be commutative). The algebra $ A $ is called Artin-Schelter Gorenstein* of dimension $ d $ if ...
Cranium Clamp's user avatar
3 votes
1 answer
231 views

Motivational distinctions between max and min conventions in tropical geometry

I am aware that algebraically, there is no real distinction between the tropical semirings $A = (\mathbb{R} \cup \{ \infty \}, \text{min}, \infty, +, 0)$ $B = (\mathbb{R} \cup \{ - \infty \}, \text{...
mijucik's user avatar
  • 177
6 votes
1 answer
157 views

Constructing countable threelds of finite dimension

A threeld is a generalization of a field, with three operations, such that the $F$ is a field with respect to the first (outer) and second (middle) operations (call it the outer field), and $F\...
Thomas's user avatar
  • 2,691
2 votes
1 answer
72 views

Is uniform dimension monotonic in quotients when there is a unique indecomposable injective?

The notion of uniform or Goldie dimension is something I’ve only seen discussed for categories of modules, but I believe the theory works the same way in any Grothendieck category $\mathcal C$. Recall ...
Tim Campion's user avatar
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