Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups.

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On finite Uniform (Goldie) dimensions

1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions? 2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to $M$...
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99 views

Systematic treatment of folding and valued graphs

I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed ...
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0answers
65 views

Structure theorem for non connected graded Hopf algebras

Let $V^{\bullet}$ be a graded vector space over a field of char $0$. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected cocommutative Hopf algebra and in ...
12
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4answers
708 views

Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product $A\...
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66 views

A property of minimal prime ideals in rings with finite chromatic number

Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...
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1answer
189 views

How “nondegenerate” are amalgamated free products of C*-algebras?

In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...
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1answer
134 views

An example of an Azumaya algebra that isn't free over its centre

Azumaya originally defined an Azumaya algebra (which he called a proper maximally central algebra) to be an algebra A which is a free module of finite rank over its centre Z such that the natural map $...
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35 views

finite orbits of transformations of the rational function field

Let $K : = k(x_1, x_2, \cdots, x_t)$ be the rational function field and consider the transformation $\tau$ of $K$ defined by $u \mapsto \frac{\alpha(u) + b}{a}$, where $a \ne 0 , b \in k$ and $\alpha$ ...
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1answer
409 views

A group algebra isomorphism problem

What are the groups $G$ and fields $\Bbb K$ for which $\Bbb K[G]\cong\Bbb K^{|G|}$ holds? For example $\Bbb R[\Bbb F_2^n]\cong\Bbb R^{2^n}$ holds.
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86 views

$T$-nilpotent ideals

Recall that a subset $I$ of a ring $R$ is left (resp., right) $T$-nilpotent in case for every sequence $$a_1,a_2,\cdots $$ in $I$ there is an $n$ such that $a_1\cdots a_n=0$ (resp., $a_n\cdots a_1=0$)....
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68 views

characterization of strong nilpotent elements

I have to referee a paper not really in my field and need some answers concerning the prime radical of a ring and nilpotent ideals. The definition of a strong nilpotent element already have appeared ...
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82 views

Cosemi-simple FRT Hopf Algebras

This question is a somewhere similar variation on an old question of mine. Given an FRT-algebra, which is to say, roughly, that it can be constructed from an R-matrix in the Yang--Baxter sense (see ...
2
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2answers
197 views

Central division algebras and splitting fields

Let $K$ be a field and $D$ be a central division algebra over $K$ of degree $n$. Suppose that $L\subset D$ is a maximal subfield, so that $[L:K]=n$. Then we know that $L$ is a splitting field, so ...
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1answer
128 views

Tensor products of simple modules over algebras [closed]

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules. We can regard $M\otimes_{\mathbb C}N$ as $A\otimes_{\mathbb C} B$-modules in natural ...
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782 views

Greatly expanded new edition of a Bourbaki chapter on algebra?

Recently I discovered by accident that Bourbaki issued in 2012 a radically expanded version of their 1958 Chapter 8 Modules et anneaux semi-simples (like other chapters, initially in French) within ...
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1answer
141 views

Are all ideals I in the ring of smooth functions on a compact manifold M equal to a set of smooth functions that vanish in $Z \subset M$?

Since $M$ is compact, we know that maximal ideals are $m_x$, the set of functions vanishing in $ x \in M$. Thus by Zorn's Lemma we also have that $I$ must sit inside such a $m_x$ for some $x \in M$. ...
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198 views

Is always the ratio (number of commuting pairs of elements in a ring or a Lie algebra)/(the size of the ring or the Lie algebra) integer?

(1) Is there a finite nilpotent ring $R$ such that the ratio $$c(R)=\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$ is not integer? Edit 1: The nilpotent condition is put later. Edit/Answer: ...
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117 views

Dimension of the center of a subalgebra of a simple algebra

Let $F$ be a field. Let $A$ be a simple (associative unital) $F$-algebra with center reduced to $F$. Let $B$ be a $F$-subalgebra of $A$; assume that $A$ is can be generated as left $B$-module by $n$ ...
7
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2answers
230 views

What is the “quaternionic” super Brauer group?

In addition to the two reasonably well-known categories $\mathrm{SuperVect}_{\mathbb R}$ and $\mathrm{SuperVect}_{\mathbb C}$ of real and complex super vector spaces, each of which is monoidally ...
2
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1answer
107 views

When do all annihilators of primitive idempotents intersect in {0}?

maybe this is silly but: for which class of rings (or commutative rings) R may I write An element a of R is zero iff for every primitive idempotent e, ea is zero ? That is, primitive idempotents "...
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2answers
83 views

Simultaneous special orthogonal similarity problem

Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that $$A=...
2
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1answer
158 views

Division and multiplication that preserve Euclidean norms

I am looking for ways to define $$\frac{1}{x}\in \mathbb{R}^n\quad \quad and\quad \quad x\cdot y\in \mathbb{R}^n ,$$ where $x,y\in \mathbb{R}^n$ such that $$\left\|\frac{1}{x}\right\|=\frac{1}{\|...
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1answer
44 views

Vanishing ideal of a finite set of points does not have expected amount of cones in Gröbner fan

I am reading the paper A Gröbner fan method for biochemical network modeling. In Chapter 4.3 (i.stack.imgur.com/h2O8B.png) they calculate the vanishing ideal of some tuples (input points of Series 1-...
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1answer
119 views

Homological characterisation of standardly stratified algebras using Ext

Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as ...
3
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0answers
279 views

A relation between a ring with its polynomial ring

Let $\{f_i(x)\}_{i\in I}$ be a subset of $R[x]$ where $R[x]$ is the polynomial ring of $R$(a commutative ring with identity). If the ring $R/\langle f_i^2(n)-f_i(n)\rangle_{i\in I, n\in A}$, for every ...
4
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1answer
108 views

Hopf-Galois Structure Maps

A Hopf-Galois extension of commutative rings, as defined by Montgomery here, is a morphism of commutative rings $\phi:A\to B$ with a Hopf-algebra $H$ coacting on $B$ by a ring map $c:B\to B\otimes H$ ...
16
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1answer
245 views

Reference request: Morita bicategory

I have two closely related questions: Who first recognized that Morita equivalence was equivalence in the bicategory of Rings, Bimodules, and Intertwiners? I've heard this bicategory called the "...
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0answers
92 views

A question about the associative classes of parabolic subgroups

Let $\mathbb{A}$ denote the adele group of $\mathbb{Q}$. Suppose that $P_1$ and $P_2$ are parabolic subgroups of a reductive algebraic group $G$, and consider their Langlands decomposition $$ P(\...
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215 views

Looking for a reference in commutative algebra

I need "I.G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973) 23–43." in my research, but it seems to be very old and rare. Does anyone know a site for ...
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114 views

The existence of zero-divisors in the universal enveloping algebra of an infinite-dimensional Lie algebra

The intuition for this problem comes from $\S$17 Exercise 1 Humphreys' Introduction to Lie Algebras and Representation Theory which essentially asks us to use PBW in order to prove that if a Lie ...
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3answers
144 views

Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?

Let $M=\begin{pmatrix} \begin{array}{cccccccc} 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\ 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\ 1 & 1 & 0 & 0 & ...
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1answer
90 views

Simple bimodule over matrix ring [closed]

Let given not trivial simple $R$- $R$ bimodule $M$, where $R$ - $n\times n$ matrix algebra over field $\mathbf{F}$. Is it true that $M$ is uniquely defined?
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45 views

Calculation of minimal right $\operatorname{add}(M)$-approximations

given a finite dimensional quiver algebra $A$ and a generator $M$ with $\operatorname{Ext}^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f\colon M_1 \...
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1answer
215 views

Definition of a normed ring

A normed ring "should" be a monoid object in the monoidal category of normed abelian groups. There are (at least) two choices of morphisms of normed groups, namely bounded or short homomorphisms, ...
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119 views

The normalizer problem for group rings

I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = $\cal{U}$$(\...
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59 views

Beaumont - Pierce Principal theorem

In book $\text{R. Göbel,P. Hill, Wolfgang "Abelian Group Theory and Related Topics"}$, I found next Beaumont -Pierce Principal theorem: Any torsion-free ring $R$ of finte rank is quasi-equal to $S\...
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180 views

Basic question about power series and complete group algebras

This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange. Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...
3
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84 views

Freeness of a matrix semigroup

Motivated by some questions in the dimension theory of self-affine sets, a colleague and I are interested in the freeness (or otherwise) of the subsemigroup of $SL_\pm(2,\mathbb{R})$ generated by the ...
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2answers
135 views

Can powers of a maximal ideal stabilize without vanishing?

Let $A$ be a local ring with maximal ideal $m$. Suppose that there exists some positive integer $k$ such that $m^k = m^{k+1}$. Is necessarily $m^k = 0$ ? If $m$ is finitely generated, this follows ...
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0answers
33 views

Tensor of Relative Bases

Suppose $U,V,W$ are the cyclotomic fields $\mathbb{Q}(\zeta_{m_3})$, $\mathbb{Q}_(\zeta_{m_2})$, and $\mathbb{Q}(\zeta_{m_1})$ respectively, where $m_1 \mid m_2 \mid m_3$ so that $U/V/W$ is a tower of ...
3
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2answers
112 views

Covering derivations of a quotient algebra

Let $(\mathcal{A},+,·)$ an algebra and $\mathcal{I}$ an ideal of $\mathcal{A}$. Is easy to check that if $D\in Der(\mathcal{A})$ with $D(\mathcal{I})\subseteq I$ then $D$ induces a derivation $D_I$ ...
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48 views

Injective Dimension of a quotient of the quantum plane

I am wondering what the injective dimension of the following ring is: $$\frac{A}{(ax, bx)}$$ where $A$ is the so-called quantum plane $k\langle x, y \rangle/(xy + yx)$ with $k$ a field, and $a, b$ are ...
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1answer
96 views

Symmetric algebras of given dimension

Fix an algebraically closed field $F$. Are there only finitely many symmetric algebras with unit over $F$ of a given finite dimension (up to isomorphism)? By symmetric I mean a Frobenius algebra where ...
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2answers
156 views

Rings all of whose torsion modules are cyclic

Let us call a (possibly non-commutative) ring $R$ "very good" if every finitely generated torsion left $R$-module is cyclic. Here is an example of such a ring: Let $k=\mathbb{C}((t))$ and let $R=k[\...
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87 views

Conjugacy classes of involutions in Kac-Moody groups

Let $A=(a_{s,s'})_{s,s'\in S}$ be a generalized Cartan matrix. Let $G=G(A)$ be the corresponding simply connected complex Kac-Moody group with Cartan subgroup $H$ and Weyl group $W$ acting on $H$. ...
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68 views

Embedding of fields in central simple algebras over number fields

Let $K$ be a non-real CM number field of degree $2d$, with maximal totally real subfield $K_0$, and let $A$ be a central simple algebra over $K$, so that $A\simeq M_n(E)$, the $n\times n$ matrix ring ...
2
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1answer
158 views

Annihilator of minimal prime ideal in a commutative Noetherian ring

Let $M$ be an $R$-module of finite length and $N$ a maximal submodule of $M.$ Is there an element $m$ in $M$ such that $m(N:M)=0$? It is a generalization of this result: In a Notherian ring $...
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162 views

Terminology for vanishing of Hochschild homology with symmetric coefficients?

In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then it should hopefully be understood by most readers as saying $H^n(A,M)=...
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28 views

Projectivity of a faithfully balanced self-orthogonal bimodule

Let $_RT_S$ be a faithully balanced self-orthogonal bimodule over a pair of noncommutative rings $(R,S)$, if $_RT$ is projective as a left $R$-module, can we say $T_S$ is also projective as a right $S$...
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179 views

Name of some commutative ring akin to $p$-adics

I need help in identifying the naming convention of some commutative ring described below. Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ...