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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Reference request: Left $R/k$-modules

In the paper titled: On the module of differentials of a noncommutative algebra and symmetric biderivations of a semiprime algebra I found the following definition: Let $k$ be a commutative ring with ...
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Conjectures inspired in the context of Casas-Alvero conjecture, via the logarithmic derivative of derivatives of a polynomial

In the post (cross-posted in Mathematics Stack Exchange with identificator MSE 4244256 and same title) we assume that $P(x)=a_0+a_{1}x+\ldots+a_{n-1}x^{n-1}+a_{n}x^n$ is a polynomial of degree $1<\...
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3 votes
1 answer
161 views

Geometric interpretation of shuffle product

Let $A=k\mathbb \Pi$ be the group algebra of an abelian group $\Pi$ and let $B(A)=\bigoplus_{k=0}^\infty\,B^k(A)$ be the unnormalized bar complex of $A$ with generators $[a_0,\dots,a_k] \in B^k(A)=A^{\...
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Modules with special properties

Let $A$ be a finite dimensional algebra and $M$ an indecomposable (right) module with the property that every nilpotent element of $End_A(M)$ annihilates the socle $soc_A(M)$ of $M$. Note that $M$ is ...
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A "semi-genetic" definition of addition and multiplication in the field $\operatorname{On}_p$?

Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$. Recursive definition of addition: $$x \oplus y := ((x+y) \...
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5 votes
3 answers
228 views

Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

Let $M $ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$ that has at least one eigenvalue of absolute value strictly bigger than $1$. What are the finite index subgroups $H$ of $\mathbb{Z}^2$ such ...
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Polyvector fields and the HKR isomorphism

I am looking for references that explain in detail the calculus of poly vector fields acting on differential forms (Gerstenhaber algebra structure) in the case of smooth commutative algebra. In ...
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Grothendieck's relative point of view and Yoneda lemma

I asked this question on M.SE, but didn't get any answers. Occasionally I hear people saying that one of Grothendieck's big insights was that often when interested in an object $X$ it's better to ...
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In $\mathbb{Z}[G]$, $G\cong \mathbb{Z}^r$, does $f\cdot g\geq 0$ imply $f\geq0$?

Let $G=\mathbb{Z}^r$ be a free abelian group, and $\mathbb{Z}[G]$ be the group ring of $G$. Define a partial ordering $\leq$ on $\mathbb{Z}[G]$ by $$\sum_{g\in G}n_g[g]\leq\sum_{g\in G}n'_g[g]\iff n_g\...
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What is mean of action of group on tower?

What is mean of action of group on tower ?in the page 17 of following paper (action on teichmuller tower ) In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class ...
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Generalization of "Lagrange interpolation" over non-division rings

The theorem below is from pages 4 and 5 in Singmaster - On polynomial functions $\pmod m$ (Theorem 10) on polynomials in $\mathbb{Z}_m[x]$. Let $f$ be a polynomial function $\pmod{m}$. Then $f$ has a ...
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Clarification about theorem on vanishing polynomials

The theorem below is from page 3 in the this paper on polynomials in $\mathbb{Z}_m[x]$. Let $F$ be a polynomial in $\mathbb{Z}_m[X]$. Then $f \equiv 0$ iff $$F \equiv F_nS_n + \sum_{k=0}^{n-1}a_k(m/(...
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Descending chain condition on finitely generated (left/right) ideals and coherent rings

Does anyone know of any interesting results on the descending chain condition for finitely generated (left/right) ideals in a unital ring, especially for coherent rings. In particular are there any ...
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Submodule of matrix space is free

Let $\mathcal{H}$ be an infinite dimensional complex inner product space, and denote by $\mathcal{H}^{n \times n} = \mathcal{H} \otimes_{\mathbb{C}} \mathbb{C}^{n \times n}$ the corresponding matrix ...
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2 votes
1 answer
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The universal envelope U(L) is a PI-algebra iff L is abelian

Let $k$ be a field with $\operatorname{char} k = 0$. Let $L$ be a Lie $k$-algebra. Then the universal envelope $U(L)$ is a PI-algebra iff $L$ is abelian. Remark: PI-algebra means polynomial identity ...
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Writing $1-xyzw$ as a sum of squares

Can you write $1 - xyzw$ in the form $p + q (1 - x^{2}-y^{2}-z^{2}-w^{2})$ where $p$ and $q$ are polynomials that are of the form $\sum g_{i}^{2}$ where $g_{i}$ $\in$ $\mathbb{R}[x,y,z,w]$? For ...
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2 votes
0 answers
127 views

Solvability and nilpotency for Banach algebras

Do we have topological counterparts of solvability and nilpotency, which are central concepts for (finite-dimensional) Lie algebras, for infinite dimensional Banach algebras with the commutator ...
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5 votes
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66 views

Isomorphism between tensor product of exterior power spaces

Suppose that $V_1, V_2, V_3$ are finite dimensional vector spaces over $\mathbb{C}$ of dimensions $d_1, d_2, d_3$, respectively. Suppose that $V_1, V_2, V_3$ are equipped with inner products, so that ...
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  • 650
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Tucker decompositions over arbitrary fields

Given an $n$-mode tensor $\mathcal{T}\in\mathbb{R}^{d_1\times\dotsb\times d_n}$, there exists a Tucker decomposition of $\mathcal T$ of the form $$\mathcal{T} = \mathcal{X}\times_1 W_1\times_2\dotsb\...
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0 answers
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How to compute the quotient and localization of the monoid algebra $kG$ for a field $k$

I am given that $k$ is a field and $G$ is the monoid consisting of all monomials $X^iY^j$, where $j$ is between $0$ and $3i$. I am trying to compute the quotient of the monoid algebra $kG$ by the ...
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3-cocycles on outer automorphism groups

Given a group $G$, the outer automorphism group $Out(G)$ acts on the center by $Z(G)$ by lifting an outer automorphism to an actual automorphism and evaluating this on elements of $Z(G)$. What is ...
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5 votes
1 answer
299 views

About the structure of unit groups appearing in number theory

I think the following statement is not true in the general situations, but consider it: $R$ is a ring, $\mathfrak{p}$ is a prime ideal, then the unit group of $\dfrac{R}{\mathfrak{p}^nR}$ is ...
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-1 votes
1 answer
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Injection vs. bijection between $V^\star \otimes V^\star$ and $(V \otimes V)^\star$ [closed]

It is a well-known fact that, if $V$ is a vector space over a field $k$, then $V^\star \otimes V^\star$ embeds into $(V \otimes V)^\star$. It turns out to be an isomorphism when $V$ is a finite-...
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3 votes
1 answer
210 views

When are simple holonomic D-modules of the form $\mathcal{D}/\mathcal{D}L$?

Let $\mathcal{D}=\mathcal{D}_X$ be the sheaf of rings of differential operators on a smooth algebraic curve $X$. Since $\dim X=1$, the D-modules of the form $\mathcal{D}/\mathcal{D}L$ are necessarily ...
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3 votes
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Are there (co-)homological obstructions to the extendability of a homomorphism?

Let $k$ be a commutative ring and $A \subset B$ an extension of $k$-algebras. Can we associate to this extension a (co-)chain complex so that its (co-)homology $H$ allows for statement such as "...
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5 votes
1 answer
366 views

Ring of continuous functions is a Jacobson ring

Let $X$ be an infinite discrete topological space. Is $$C_b(X)=\{ f \colon X \to \mathbb{R} \text{ bounded }\}$$ a Jacobson ring ?
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7 votes
1 answer
367 views

Factorization of an irreducible polynomial in the field extension it defines

In field theory, the following fact is used in the construction of splitting fields: Given a field $F$ and an irreducible polynomial $f \in F[x]$, the quotient $F[\alpha]/(f(\alpha))$ is a field ...
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3 votes
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How to go from primitive idempotents in $\text{End}_A(M)$ to primitive idempotents in $A$?

Let $K$ be a large enough finite field, let $A$ be a finite-dimensional $K$-algebra. Moreover, let $M$ be a finitely generated $A$-module and let $M = M_1\oplus ... \oplus M_n$ be a decomposition of $...
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7 votes
2 answers
377 views

When is the rank of $AB+BA$ equal to one?

For two arbitrary matrices $A$ and $B$, are there any known conditions for the rank of $AB+BA$ to be equal to one?
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8 votes
2 answers
184 views

Lifting isomorphisms between linear categories

Let $C$ be a $\mathbb{Z}$-linear category, such that $C(x,y)$ is a free abelian group with finite rank, for every $x,y\in\mathrm {Ob}(C)$. Given a commutative ring with identity $R$, let $RC$ denote ...
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9 votes
1 answer
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Examples of non-self-induced algebras

Let $A$ be a (possibly non-unital) algebra over $\mathbb C$. We say that $A$ is self-induced if the product map $m:A \otimes_A A \rightarrow A$ is an isomorphism. Here $A \otimes_A A$ is the ...
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4 votes
0 answers
68 views

Nullstellensatz for maximal left ideals of quantum plane

Let $R=\mathbb{C}\langle x,y\rangle/\langle xy=qyx\rangle$ be the quantum plane algebra. Does some sort of Nullstellensatz holds for the maximal left ideals of $R$? By this we mean all maximal left ...
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2 votes
1 answer
143 views

Primitive elements in the universal enveloping algebra of Lie superalgebra

Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
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8 votes
0 answers
230 views

Does the book "Algebra III" exist (within the Encyclopaedia of Mathematical Sciences series from Springer)?

Within the series "Encyclopaedia of Mathematical Sciences", as published by Springer, one finds the 8 volumes, namely, the volumes I, II, IV, V, VI, VII, VIII, IX but zbMath has no listing ...
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1 vote
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A finite $H$-module algebra is necessarily inner

Let $H$ be a finite-dimensional Hopf algebra over a field $k$, and $A$ an $H$-module algebra. If $A$ is finite-dimensional over $k$, is the action of $H$ on $A$ necessarily inner? If this is not true ...
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3 votes
0 answers
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Terminology question

Let $K$ be a commutative ring with unit and suppose that $R$ is a ring that is a left $K$-module satisfying $c(rs)=(cr)s$ for all $c\in K$ and $r,s\in R$. We do not require that $r(cs)=c(rs)$ and so $...
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0 votes
0 answers
154 views

The coevaluation map for a projective module and its dual

$\DeclareMathOperator\coev{coev}$Let $R$ be a noncommutative ring and let $P$ be a bimodule over $R$, that is finitely generated and projective as a left module. It is "well-known" that any ...
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2 votes
0 answers
59 views

Prime elements in integrally closed extension of domains

Let $R \subseteq S$ be an extension of integral domains such that $R$ is integrally closed in $S$. Let $P$ be a prime ideal of $R$. Is $PS$ always a prime ideal of $S$? For classical examples of ...
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2 votes
0 answers
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Non-associative algebras and determinant over 3 by 3 matrices

I have a non associative algebra $A$ that is not unital. And I have three by three matrices $X$ with coefficients over $A$ with a matrix product $*$. I'd like to define something like the determinant ...
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7 votes
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Identity for the associator involving a third root of unity

This is a reference request. I came across the class of nonassociative algebras satisfying the following identity: $$ (a,b,c)+\omega(b,c,a)+\omega^2(c,a,b)=0. $$ Here: by an "algebra" I mean a ...
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2 votes
0 answers
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Local rings vs local Banach algebras

$\DeclareMathOperator\rad{rad}$Suppose $R$ is a ring and let $R^{-1}$ be the set of invertibles in $R$. Then we know there are examples of noncommutative rings st $R = R^{-} \cup \rad R$, where $\rad ...
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2 votes
1 answer
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Nondegenerate pairings versus perfect pairings for finitely generated projective modules

Let $R$ be a (not necessarily commutative) ring, $M$ a left $R$-module, and $N$ a right $R$-module. We say that a pairing $$ \langle -,-\rangle:M \otimes_R N \to R $$ is non-degenerate if, for all $n \...
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3 votes
1 answer
84 views

Quiver algebras from semirings and posets as semirings

A semiring is a nonempty set $S$ such two binary operations + and * making S into a semigroup with + and * and such that a*(b+c)=ab+ac and (b+c)a=ba+c*a for all a,b,c in S. Assume in the following ...
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  • 21.8k
10 votes
2 answers
662 views

The maximal subset of a finite field where the sum of any subset is non-zero

Given a finite field $\mathbb{F}_q$ with $q=p^m$ where $p$ is the characteristic. For any subset $S=\{a_1,\dots,a_n\}$ of $\mathbb{F}_q$, if any partial sum (i.e. the sum of elements in a non-empty ...
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9 votes
1 answer
321 views

Hopf algebra with a non-invertible antipode

What is an example of a Hopf algebra with a non-invertible antipode?
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12 votes
0 answers
327 views

Why is it so hard to give examples of differentially closed fields?

The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...
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  • 21.7k
5 votes
1 answer
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Finding non-inner derivations of simple $\mathbb Q$-algebras

What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)? I'm under the impression that when ...
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  • 1,127
6 votes
2 answers
221 views

Abelian groups such that $A \cong \mathrm{End}(A)$ and "complete rings"

Motivation: for any ring $R$ there is the natural monomorphism $\mathrm{in} \colon R \to \mathrm{End}(R_{add}): r \mapsto (x \mapsto rx)$, where $R_{add}$ is an additive abelian group ( rings are ...
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6 votes
1 answer
351 views

Is Solèr’s theorem true in constructive mathematics?

Solèr’s theorem says that for every star division ring $R$ and every $R$-module $H$ with an orthomodular Hermitian form $\langle (-),(-) \rangle:H \times H \to R$ such that there exists an infinite ...
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17 votes
1 answer
550 views

Is Hurwitz's theorem true in constructive mathematics?

Hurwitz's theorem says that the only division composition algebras over the real numbers $\mathbb{R}$ are the real numbers themselves $\mathbb{R}$, the complex numbers $\mathbb{C}$, the quaternions $\...
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