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.
3,342
questions
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Real exponentiation in the quotients of rings of continuous functions by prime ideals
Consider the ring $C = C(X) = C(X; \mathbb{R})$ of continuous functions $f:X\to \mathbb{R}$ where $X$ is a Tychonoff space. This is naturally a lattice ordered ring by setting $f\geq 0$ iff $f(x)\geq ...
2
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48
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Quasi-isomorphisms of P-algebras
In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...
4
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1
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111
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Examples of Noetherian integral group ring
I need to study the integral group ring of the fundamental group of a manifold. My knowledge of group and ring theory is very limited. I am looking for some examples of groups $G$ for which $\Bbb ZG$ ...
1
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1
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97
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Particular example of a quadratic extension of a nonunital ring
I want to construct a concrete non-unital ring $R$ with the following properties:
$R$ is a noncommutative non-unital ring with a right unite $r$ i.e $t.r=t$ for any $t\in R$.
$S\subset R$ is a ...
19
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2
answers
649
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Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?
With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?
Here, by "trivial examples" ...
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Extending the proof of Maschke's Theorem from finite groups to algebras
In the theory of representations of a finite group there is Maschke's Theorem that any finite-dimensional representation of a finite group $G$ can be decomposed into a direct sum of irreducible ...
2
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1
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188
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Concrete examples of derived categories
What examples of abelian categories $\mathcal{A}$ are there such that the derived category $\mathcal{D}(\mathcal{A})$ can be described concretely? For example, is there a concrete way of describing $\...
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96
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Representation of anti-commuting matrices in $\mathbb{C}^{2}$
This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem.
The basic question is the following. Let $V$ be a finite-...
6
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1
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165
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A name for semigroups in which left and right principal ideals coincide
Is there any standard name for semigroups $S$ in which $xS=Sx$ for all $x\in S$?
Examples of such semigroups are commutative semigroups and Clifford inverse semigroups.
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When does the sum of squares/cubes of polynomials over finite field have less than maximum degree?
Given polynomials $p_1(x), p_2(x), \dots p_m(x) \in \mathbb{F}_p[x]/\langle x^p-x\rangle$ where $p$ is a prime, when does $\sum_{i=1}^m p^2_i(x)$ have degree $< p-1$? What about $\sum_{i=1}^m p^3_i(...
2
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n-ary (polyadic) group "defined for tuples"
Consider an n-ary (polyadic) group, which is a generalization of the "usual" ("binary") groups to n-tuples. The definition requires appropriate standard generalizations of ...
4
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Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$
Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread.
Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
1
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1
answer
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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 ...
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2
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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 $\|\cdot\|$ ...
2
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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 $\...
2
votes
2
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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 ...
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0
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48
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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{...
4
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162
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Infinite-dimensional, non-unital Frobenius algebras
A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space over some field, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital ...
5
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2
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386
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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 ...
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0
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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 $...
7
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1
answer
276
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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 $...
2
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1
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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?
3
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1
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233
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Locate the paper "Extensions of general algebras" by Eilenberg
I'm looking for the paper
Extensions of general algebras, S. Eilenberg, Ann. Soc. Polon. Math. 21, (1948). 125–134 (MR0026647 in Mathscinet)
It is supposedly hosted at the Digital Repository of ...
11
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2
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523
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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 ...
17
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0
answers
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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*}
[...
2
votes
0
answers
92
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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 ...
18
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3
answers
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How do I apply the Boolean Prime Ideal Theorem?
I have become aware of an amazing phenomenon from a myriad of questions and answers here on MathOverflow: many of the results that I would typically prove using the Axiom of Choice can actually be ...
3
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1
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121
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Are there examples of brace algebras that are not operads?
The most typical example of a brace algebra is the brace algebra structure on the Hochschild complex of an associative algebra. This is a particular case of the following construction applied to the ...
11
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0
answers
471
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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 ...
1
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1
answer
345
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Reference request: a cousin to the log semiring
Let $f$ be strictly increasing on $\mathbb{R}$. Then $x \oplus y := f^{-1}(f(x)+f(y))$ gives rise to a strict symmetric monoidal ($\Rightarrow$ commutative monoid) structure on $(\mathbb{R},\ge)$ with ...
6
votes
1
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375
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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].
...
1
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0
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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 $\Delta x$...
3
votes
0
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101
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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 ...
0
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0
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55
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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 ...
2
votes
0
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144
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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 ...
9
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1
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195
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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 $T$?
4
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0
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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 ...
6
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1
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474
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Normal form for terms in language with two ring structures
Suppose I have two different ring structures on the same domain $\langle R,+,\cdot,0,1\rangle$, $\langle R,\oplus,\otimes,\bar 0,\bar 1\rangle$ and I throw the structures together into a single common ...
2
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1
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375
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Extending a complete lattice to get a "nice" Boolean lattice
Suppose we have a complete lattice. Which additional axioms (e.g. distributivity axioms) are needed to obtain a Boolean lattice in which $\text{complement}(a) = \text{lub}\{b \;|\; b \wedge a = \text{...
3
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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, ...
2
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1
answer
65
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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 \...
0
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0
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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 ...
0
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0
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92
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Isomorphism in division algebras [closed]
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 ...
12
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2
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751
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Description of p-adics tensor the reals
What is $\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathbb{R}$ equivalent to?
where $\mathbb{Z}_p$ are the p-adic integers.
I am specially interested in the case $p=2$.
Do know that $\mathbb{Z}_p\otimes_{\...
9
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2
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563
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Birkhoff's completeness theorem put into practice
Birkhoff's completeness theorem (see here, Theorem 14.19) states that an equation which is true in all models of an algebraic theory can be proven in equational logic.
Question. Does the proof of ...
2
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1
answer
139
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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 ...
7
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0
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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
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0
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86
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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).
...
24
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3
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Why is a ring called a "ring"?
Why is a ring called "ring" (or Zahlring in German)? There seems to (naive) me nothing more ring-like to a ring than there is to a group or a field. I am particularly interested to learn why the ...
10
votes
2
answers
485
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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 $F=E$?