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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A question on symmetric functions

Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...
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Does a given bound quiver algebra admit an algebra analogous to the preprojective algebra of a path algebra?

Let $Q=(Q_0,Q_1,s,e)$ be a finite quiver. In the formal construction of the preprojective algebra associated to $Q$, we set $\ Q^*_1=\{\alpha^*:y\rightarrow x| \alpha\in Q_1, \alpha:x\rightarrow y \}...
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Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$

I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...
Ali Taghavi's user avatar
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Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...
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Centralizers of elements in free group algebras

Let $A$ be a group algebra of a free group, and $x \in A$. What is the centralizer of $x$? Is there something like Bergman's theorem for free associative algebras?
Olga Kharlampovich's user avatar
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Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$. Q1) Is it ...
Subhajit Jana's user avatar
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A generalization of real characters on a group

Yesterday I understood that I can't live without this construction: Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps $\varphi_k:...
Sergei Akbarov's user avatar
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Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
yang's user avatar
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Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients. $$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$ Monomials $x^k$ are mapped to $n ...
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Sum of projective submodules of a projective over a semihereditary ring

Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a ...
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Chinese remainder theorem

For non-commutative rings, we have this generalization of the Chinese remainder theorem (CRT). I wonder if there is another statement involving only left or right ideals; do you know any?
Exodd's user avatar
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Injectivity criterion for surjective coalgebra maps: does it hold in full generality?

Let $\mathbf{k}$ be a commutative ring. Let $C$ be a filtered $\mathbf{k}$-coalgebra. This means a $\mathbf{k}$-coalgebra equipped with an increasing $\mathbf{k}$-module filtration $C^0 \subseteq C^1 \...
darij grinberg's user avatar
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What is the composition in SesquiAlg?

To motivate my question, I will describe a famous 2-category. First, there is the 1-category $\text{Vect}$ of vector spaces (over some fixed field). This category has a tensor product $\otimes$, and ...
Theo Johnson-Freyd's user avatar
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On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix. Suppose there exists ...
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Homological dimension of completed path algebras.

Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations. Is it true that the I-adic completion of A has finite homological dimension?
Andre 's user avatar
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Cocontinuous monadic functors

Some forgetful functors of algebraic categories preserve colimits, but most do not. In order to understand this phenomen in general, I have classified cocontinuous monadic functors whose target is an ...
Martin Brandenburg's user avatar
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Elementary polynomial-free proofs of fundamental theorem of Galois theory?

I am looking for simple proofs that show the correspondence between intermediate fields in a field extension and subgroups of the Galois group. I'm happy for everything to be subfields of $\mathbb{C}$....
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Inverse limit of graded rings

Let $(I,\le)$ be a directed set and let $(\rho^{\beta\alpha}: R^\beta \to R^\alpha)_{\alpha \le \beta}$ be an $I$-directed system of $\mathbb{Z}$-graded rings whose multiplication is denoted by $$\...
Ralph's user avatar
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A reference on semisimple linear algebra

Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings? In fact this question is a ...
Alexander Shamov's user avatar
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integral versus adjoint action on Hopf algebra

Suppose that $H$ is a finite dimensional Hopf algebra (with counit $\varepsilon$) and $T$ is a non zero right integral of $H^{\star}$ (the dual Hopf algebra). Let $ad_h$ be the adjoint action on $H$, ...
user25332's user avatar
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Roots of unity in algebraic K-theory

For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit. For $...
Craig Westerland's user avatar
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Ring properties which can be checked on sufficiently many Ore localizations

Let $R$ be a non-commutative Ore domain, and let $R[S_i^{-1}]$ be a finite set of Ore localizations. The absence of a geometric theory means there is no obvious candidate for when this set of ...
Greg Muller's user avatar
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Unitary unit conjecture for group rings

The famous "unit conjecture" for group rings states that all units of a group ring $K[G]$ are trivial for a field $K$ and a torison-free group $G$. We are far away from solving the conjecture (See e.g....
Joerg Sixt's user avatar
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Monomial-type ideals in polynomial rings

Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ...
Timothy Wagner's user avatar
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localization at central maximal ideals

If M is a maximal ideal of Z(R), the center of a ring with identity R, and R_M, the localization of R at M, is a commutative field, what can we say about R? My guess is that we can's say that much ...
iravan's user avatar
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is there a notion of weakly noetherian?

A left module M over a ring is finitely presented iff the functor Hom(M,_) preserves filtered inductive limit. Meanwhile, if M is finitely generated, then for every filtered system { N_i } of left ...
Carl Weisman's user avatar
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Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups

In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete ...
Mikola's user avatar
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Examples of noncommutative Bezout domains

I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...
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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 ...
Amanuel Jissa's user avatar
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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
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A basis for the 0-Hecke ring

Let $(W,S)$ be a Coxeter system of type $A_n$, with $$S=\{s_1,\ldots,s_n\}$$ satisfying the usual relations, and let $R=\mathbb{Z}[x_1,\ldots,x_{n+1}]$ be a polynomial ring. $W$ acts on $R$ by ...
Matt Samuel's user avatar
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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 ...
Qwert Otto's user avatar
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Definitions of torch ring

Well, I find myself in a tangled web of references, definitions and doubt. How do I get myself into these things? Get ready for a rollercoaster of definitions. An FGC ring is a commutative ring whose ...
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Road map for learning cluster algebras

I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
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Real forms of complex Lie algebras with specified semisimple part

Let $\mathfrak g$ be a complex Lie algebra. If $\mathfrak g$ is semisimple then its real forms are completely understood. In particular there is a compact real form of $\mathfrak g$, which is unique ...
Claudio Gorodski's user avatar
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On skew monoid rings and skew ordered series rings

To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...
Salvo Tringali's user avatar
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Characterizing atomicity in a commutative domain

In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ...
Salvo Tringali's user avatar
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Does there exist a finitely generated, torsion group $G$ with a residually finite ascending HNN extension?

I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can provide me some insight. Let $G$ be a group with an injective endomorphism $\phi$...
ghc1997's user avatar
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Non-Desarguesian finite projective planes with ≤3 (non-collinear) chosen points, and coordinatisation

It is well-known that an arbitrary projective plane can have very different symmetry group to a field plane. In particular, the symmetries are not transitive on the set of fundamental quadrangles. ...
David Roberts's user avatar
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7D simple Lie algebras over $\mathbb{F}_3$

Up to isomorphism, what are all the seven-dimensional simple Lie algebras over the field with three elements?
Daniel Sebald's user avatar
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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 ...
darkl's user avatar
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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 ...
user498029's user avatar
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A non-commutative, left duo ring whose only unit is the identity

Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a left duo ring if $aR \subseteq Ra$ for every $a \in R$. Question. Is there a non-commutative, left ...
Salvo Tringali's user avatar
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105 views

Conilpotent coalgebras as pushouts of trivial coalgebras

Let $K$ be a field and $C$ a non-counital conilpotent coassociative coalgebra over $K$ whose underlying $K$-vector space is finite dimensional. Question: Can one obtain $C$ by iterately taking ...
Hadrian Heine's user avatar
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291 views

Automorphisms of the ring of periods

The set of periods $\mathcal{P}$ introduced by Kontsevich and Zagier forms a ring, see for example https://en.m.wikipedia.org/wiki/Period_(algebraic_geometry). Moreover J. Wan introduced in 2011 in ...
Sylvain JULIEN's user avatar
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347 views

A projective module over a domain that is not faithfully flat?

Let $R$ be a (noncommutative) unital ring which is a domain and let $\mathcal{N}$ be a non-zero projective (right) module. Projectivity of course implies that $\mathcal{N}$ is flat, but does the fact ...
Tim Montegue's user avatar
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188 views

Does the tropical semiring admit a universal property?

Each of the rings $\mathbf{Q}$, $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathbf{R}$ admits a universal construction: the rationals are the field of fractions of the integers: $\mathbf{Q}:=\mathrm{Frac}(\...
Emily's user avatar
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Associative rings with "big" commutative subrings

Let $A$ be an associative ring and $R\subset A$ be a commutative subring. Suppose that every element of $A$ has the form $urv$ where $r\in R$ and $u, v\in A^*$ are invertible. A basic example is $A=...
Eurydice's user avatar
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Are PD-, $\lambda$-, $\psi$-, and $\delta$-rings monoids in a monoidal category?

$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category: A monoid ...
Emily's user avatar
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Question about terminology for a class of "self-modular" mappings between rings

(In the scenario I have in mind, rings need not be unital.) The following notion has come up in some joint work that is being written up. Let $R$ and $S$ be rings, and let $D$ be a subring of $R$. Is ...
Yemon Choi's user avatar
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