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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80 views

Determinant of chain complexes

Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of ...
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+50

Do compact inverse-property loops (or just compact Moufang loops) have bi-invariant Haar measure?

So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left invariant? (And we can restrict to Moufang loops if ...
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77 views

On infinite global dimensions of "slightly non-commutative" rings

Assume $R$ is a commutative Noetherian ring of finite Krull dimension; $R'$ is a not commutative ring that contains $R$ in its center and also finitely generated as an $R$-module. If the (left) global ...
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1answer
300 views

For every ring R, is there a block-diagonal canonical form for a square matrix over R?

This question asks whether there exists an analogue of the Jordan decomposition for an arbitrary ring $R$. This analogue is not necessarily the Jordan-Chevalley decomposition, which is unnecessarily ...
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1answer
106 views

Is a certain map a quasi-isomorphism?

$\DeclareMathOperator\Hom{Hom}$Assume $F$ and $M$ are respectively right and left modules over a ring $R$ and let $I^\bullet$ be a left-bounded exact complex of $R$-$R$-bimodules. We know there is a ...
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1answer
66 views

Rings of finite uniserial type

If $R$ is a ring and $M$ an $R$-module, $M$ is uniserial if its lattice of submodules is a chain. Over an Artinian $R$, the chain will be finite. From what I understand, deciding when two uniserial ...
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1answer
147 views

Symbolic powers of a prime ideal of height one

Can someone please give me an example of a Noetherian normal local domain of dimension two such that there exists a prime ideal $P$ of height one having the property $P^{(n)}$ is not a principal ideal ...
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1answer
456 views

Under which conditions: dim(W1 + W2 + W3) = dim(W1) + dim(W2) + dim(W3) − dim(W1 ∩ W2) − dim(W2 ∩ W3) − dim(W3 ∩ W1) + dim(W1 ∩ W2 ∩ W3) [closed]

Let $V$ be a finite dimensional vector space over a field $K$, and let $W_1$, $W_2$ and $W_3$ be subspaces of $V$. By analogy with the inclusion-exclusion principle for sets, and taking into account ...
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136 views

Is it correct to use extensionality axiom in an algebraic theory? Is "extensionality theory" appropriate name for the identity theory plus this axiom?

There is something about extensionality axiom which makes debatable its use in any theory, not only in an algebraic one -- this law looks more like a definition than a statement when written like this:...
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63 views

Structure of finitely generated $\mathbb{Z}/p^n\mathbb{Z}[[S,T]]$-modules

Let $\Omega=\mathbb{Z}/p\mathbb{Z}[[S,T]]$. $\Omega$ is a commutative, Noetherian and integrally closed domain of Krull dimension 2. According to Bourbaki's commutative algebra VII $\S 4$, if $M$ is a ...
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1answer
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Auslander-Reiten quiver of quiver algebra kQ where Q is of extended dynkin type D4~

I am looking for literature about the Auslander-Reiten quiver of the quiver algebra $kQ$, where $Q$ is of extended dynkin type $\tilde{D_4}$ and $k$ is an algebraically closed field. Does somebody ...
2
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1answer
119 views

Alternativity on $A \otimes B$

I have $A$ an associative algebra and $B$ at least an alternative algebra. Is there a sufficient condition on $A$ or $B$ to have $A \otimes B$ an alternative algebra?
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1answer
156 views

Is the theory of the adjunction operation, used for algebraization of hereditarily finite set theory, algebraizable?

The operation used for algebraization of hereditarily finite set theory is named adjunction in 〈A. Tarski, S. Givant, A Formalization of Set Theory without Variables, Providence, RI: American ...
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117 views

Antipode on the dual multiplier Hopf algebra

Consider the following fragment in the proof of proposition 4.7 from the paper "An algebraic framework for group duality" by Van Daele. Here, $\omega_1, \omega_2$ are elements of the dual $\...
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Matrix decompositions as monoid isomorphisms. Ever considered before?

I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question: ...
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55 views

Nonassociative quaternion algebras

I'm interested in nonassociative central simple algebras; I've found Lee and Waterhouse's article 'Maximal Orders in Nonassociative Quaternion Algebras', and this cites a previous article by ...
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105 views

Semigroup ideals of a ring or an algebra

Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
2
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1answer
178 views

Is the number of values the sign function can take on a ring ("signedness") of any fundamental importance? Can it be predicted?

There are well-described methods of generalizing arbitrary functions to matrices in a natural way. Basically, if $A=PD_AP^{-1}$ where $D_A$ is a diagonal matrix, then $f(A)=Pf(D_A)P^{-1}$, where the ...
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62 views

When is a bounded complex of $RG$-modules contractible?

If we have a $p$-modular system $(K, \mathcal{O},k)$, let $R = k$ or $\mathcal{O}$ and $G$ a finite group. When is a bounded complex of $RG-RG$-bimodules $\Gamma$ contractible? I've seen this response ...
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61 views

Localization of the injective hull of a commutative non-Noetherian ring

Let $R$ be a commutative non-Noetherian ring and $m$ a maximal ideal. My question is whether the localization $E(R)_m$ of the injective hull $E(R)$ of $R$ is an injective $R_m$-module. This is true in ...
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2answers
909 views

A ring for which the category of left and right modules are distinct

What is an example of a ring $R$ for which the abelian category of left $R$-modules is not isomorphic to the category of right $R$-modules?
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66 views

When every principal annihilator is prime

Let $ M$ be a unit $ R$-module over commutative ring $R.$ Is there any equivalent condition that for every element $x $ of $M $ either $ann (x)=R$ or $ann (x)$ is a prime ideal of $R$? where $ann (x)=\...
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38 views

Exponential of a sum in a non-commutative graded algebra

Let $a,b$ be two elements of a graded algebra $A$ such that $\deg(a)=1$, $\deg(b)=0$ and $[a,b]\neq 0$. I would like to know whether there exits an explicit expression for the degree 1 component $$\...
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1answer
114 views

When the annihilator of each nonzero submodule is prime

Let $M$ be a fixed faithful $R$-module over integral domain $R$. Is there any equivalent condition (on $R$ or on $M $) under which the annihilator of any nonzero submodule of $M$ to be a prime ideal ...
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1answer
208 views

Covolumes of unit groups of division algebras

Let $D$ be a central division (or maybe just simple) algebra over $\mathbb{Q}$. Let $\mathcal{O} \subset \mathcal{O}_m$ be an order inside a fixed maximal order and denote by $\mathcal{O}^1$ its group ...
2
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1answer
248 views

Can you compute the Krull dimension of a subalgebra using ideals?

$\DeclareMathOperator\height{height}$Let $k$ be algebraically closed, $A$ be a $k$-algebra of finite type and $B$ a sub-$k$-algebra of $A$. Assume there exists a largest ideal $I$ of $A$ such that $I \...
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1answer
87 views

A nice/simple relationship between the Chevalley generators of $\mathfrak{sp}_n$ and the Chevally generators of $\mathfrak{sl}_n$?

The Lie algebra $\mathfrak{sl}_n$ is defined to be the trace free matrices in $M_n(\mathbb{C})$. The Lie algebra $\mathfrak{so}_n$ is defined to be the matrices $A$ in $M_n(\mathbb{R})$ satisfying $A +...
7
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1answer
298 views

This is not a tensor: tensoring abelian groups over groups

$\newcommand{\Cats}{\mathsf{Cats}}\newcommand{\MonCats}{\mathsf{MonCats}}\newcommand{\BrMonCats}{\mathsf{BrMonCats}}\newcommand{\SymMonCats}{\mathsf{SymMonCats}}\newcommand{\CMon}{\mathsf{CMon}}\...
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163 views

Coordinate ring of a flag variety

Edited: [If G here is a simply connected semismple complex algebraic group. A partial flag variety $G/P$ can be naturally embedded as a closed subset of $\prod_j \mathbb{P} (L(\omega_j)^*)$. The ...
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1answer
600 views

Existence of a finite extension of ℤ providing a finite extension of the primes

Let $R$ be a ring (possibly noncommutative with zero-divisors). A non-unit and non-zero-divisor element $r \in R$ will be called irreducible if for all $a,b \in R$ such that $r=ab$, then $a$ or $b$ is ...
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126 views

Ideal generated by a regular sequence

In Boocher and Grifo - Lower bounds on Betti numbers, in example 2.2 they say that if $R=k[x_1,\dotsc,x_n]$ is a polynomial ring and $M=R/(f_1,\dotsc,f_c)$ where $f_i$ form a regular sequence, then ...
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1answer
291 views

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 ...
9
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1answer
311 views

Is the Magnus Lie algebra of a finitely presented group finitely presented

Let $G$ be a finitely presented group and let $L(G)$ be the Magnus Lie algebra associated to the lower central series of $G$. This $L(G)$ is a graded Lie ring generated by its degree 1 piece $L_1(G) =...
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2answers
472 views

Is there any example of a Lie algebra which is not a derivation algebra?

I'm just studying Lie algebras. If $A$ is a $k$-algebra (not necessarily Lie or associative, just a bilinear law), it is straightforward to check that any derivation algebra of $A$ is a Lie algebra. I ...
3
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1answer
208 views

RIng that is flat over a subring as a right module but not as a left module

What is an example of a ring $R$ and a subring $S \subseteq R$ such that $R$ is flat as a right module but not flat as a left module. The following question is my motivation: Faithful flatness for ...
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2answers
214 views

Relation of the first Hochschild cohomology and the outer automorphism group

Let $R$ be a ring. Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite? (It is not true, by the two answers. Is it ...
3
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1answer
107 views

Hochschild cohomology of finite semisimple algebras

Let $A$ be a finite semisimple algebra over $\mathbb{k}$, a perfect field. Is true that the second Hochschild cohomology group vanishes, i.e. $$HH^2(A) = 0?$$ In order to make this question a little ...
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265 views

Near-ring spaces

$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\...
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1answer
211 views

Can you constructively prove a univariate polynomial algebra over a Jacobson ring is itself Jacobson?

Recall the Jacobson radical of a commutative ring $\mathrm J(A)=\lbrace a\in A\mid \forall b\in A:1-ba\in A^\times\rbrace$. The Jacobson radical of a quotient by an ideal $I\vartriangleleft A$ is ...
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41 views

Subalgebra of a crossed product central division algebra, generated by powers of group elements

Let $k=\mathbb C$, let $K$ be a finite extension of the field $k(X_1,X_2,X_3)$ of rational functions in $3$ variables, let $L/K$ be a finite Galois extension of commutative fields and let the Galois ...
3
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0answers
254 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 ...
4
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1answer
203 views

Is a non-degenerate finite-dimensional algebra unital?

Let $A$ be a finite-dimensional (not necessarily unital) associative algebra over the field of complex numbers $\mathbb{C}$ (but I'm also interested in more general fields). Assume the multiplication ...
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1answer
147 views

Can we say that everywhere where it makes sense $\log_0 x=0^x$? Are they equal, the function is self-inverse? If so, what is deep intuition behind it? [closed]

It makes little reason to speak about $0^x$ and $\log_0 x$ on the set of real numbers, but in matrices, it seems, the expressions coincide, for instance, $0^ \left( \begin{array}{cc} \frac{1}{2} &...
2
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1answer
73 views

Terminology for commutative ring whose Jacobson radical $J$ is nilpotent with semisimple quotient $R/J$

Is there a name for the following property of a commutative ring $R$: its Jacobson radical $J$ is nilpotent, and $R/J$ is semi-simple? (It is easily equivalent to: $R$ is a finite product of ...
4
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0answers
276 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 ...
2
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0answers
81 views

Expressing elements in Verlinde ideal in terms of generators

It is known that the level $k$ Verlinde ring of $SU(n)$ is $R(SU(n))/I_k$, where $I_k$ is the Verlinde ideal. A set of generators of $I_k$ is given by $\{V_{(k+i)L_1}:=\text{Sym}^{k+i}V_{\text{std}}| ...
2
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2answers
149 views

Module complements to rings embedded in a projective module

Let $R$ be noncommutative unital ring and $M$ a projective (right) $M$-module. Assume that $R$ embedds into $M$ as a right -module. A) If $R$ is a semisimple ring, then of course $R$ admits an $R$-...
34
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15answers
3k views

Results in linear algebra that depend on the choice of field

Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers). I am looking for a list of concepts, and results, in ...
7
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1answer
474 views

First isomorphism theorem for sets?

Let $f\colon S\to T$ be any function. There is the obvious refinement of $f$, by replacing the codomain $T$ with the image. Thus, every function factors into a surjection followed by an injection (...
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65 views

Composition of faithfully flat ring extensions

Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, ...

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