# Tagged Questions

**16**

votes

**1**answer

236 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 ...

**1**

vote

**0**answers

91 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
$$
...

**2**

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**0**answers

211 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 ...

**2**

votes

**0**answers

108 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 ...

**3**

votes

**3**answers

142 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 & ...

**-5**

votes

**1**answer

89 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?

**2**

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**0**answers

44 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 ...

**5**

votes

**1**answer

210 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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**0**answers

114 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 $\cal{U}$ = ...

**1**

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**0**answers

58 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 ...

**0**

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**0**answers

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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**0**answers

83 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 ...

**1**

vote

**2**answers

133 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 ...

**2**

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**0**answers

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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**2**answers

110 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 $D_I$ ...

**0**

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**0**answers

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 ...

**5**

votes

**1**answer

95 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 ...

**3**

votes

**2**answers

153 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 ...

**2**

votes

**0**answers

85 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 $H$.
...

**1**

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**0**answers

67 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**

votes

**1**answer

128 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 an generalization of this result:
In a Notherian ring R, all minimal prime ...

**7**

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**0**answers

160 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 ...

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**0**answers

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 ...

**0**

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**0**answers

177 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 ...

**2**

votes

**3**answers

325 views

### Subring of ring

Let given ring $R$ without zero divizors, where adittive group of $R$ with zero torsion. Let given subring $R_0\leq R$, and $p$ is prime number, such that $\forall r\in R, \exists i>0 : p^ir\in ...

**9**

votes

**1**answer

232 views

### Does it follow that any element of $J(A)$ is nilpotent?

Let $A[x]$ be the algebra of polynomials with coefficients in a $k$-algebra $A$. Assume that, for any simple $A[x]$-module $M$, we have $\text{End}_{A[x]} M = k$. Does it follow that any element of ...

**2**

votes

**1**answer

92 views

### projective module of rank one over notherian ring

Is finitely generated projective module M of rank one over regular commutative notherian ring free?
Bass (Illinois Math J, 1963) showed that in case M is nonfinitely generated, it is free. I am ...

**0**

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**0**answers

50 views

### One-sided endomorphism rings of centred bimodules

Let R be an associative unital ring. An R-bimodule M is called centred bimodule if M = R*Z(M), where Z(M)={m:rm=mr,∀r∈R}, i.e., M is generated as an R-module by the set of R-centralizing elements. ...

**0**

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105 views

### Quotient modules of polynomial rings by maximal one-sided ideal

Let $R[X]$ be a ring of polynomials over an associative unital ring $R$ which is not necessarily commutative. Let $M$ be a maximal left ideal in $R[X]$. It is easy to see that if the intersection of ...

**2**

votes

**1**answer

204 views

### Finite rank ring

Let given ring $R$ of finite rank. Is it true that for all primes $p$ large enough modules $Der_{\mathbb{Z}}(R/pR) = \{0\}$?
For every ring we define $Der_{\mathbb{Z}}(R)$ as set of linear operators ...

**19**

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**0**answers

273 views

### Topological loops vs. algebro-geometric suspension in Hochschild homology

Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...

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33 views

### connected stable rank

There is a beautiful formula by Leonid Vaserstein relating the Bass and topological stable rank of a commutative unital Banach algebra A to
that of the matrix algebra M_n(A). Is there something ...

**1**

vote

**0**answers

63 views

### notation for vector product in the space

The notation for vector (a.k.a. cross) product in $\mathbb{R}^3$ I usually see is $\times$.
However, some places use $\wedge$ instead, which IMHO creates a lot of confusion, as $\wedge$ usually is ...

**2**

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**0**answers

119 views

### For f: X -> Y -> X, what is the name of the property whereby for all x in X and y1, y2 in Y, f(f x y1)y2 = f(f x y2)y1?

My PL group has been discussing this (so we are really professional reject mathematicians, but this is more about the math behind what we are doing). Some of us called it a "generalized ...

**5**

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**2**answers

184 views

### Lefschetz Principle for semisimplicity

I think I can prove the following using the compactness of first order logic and I am wondering what a purely algebraic proof would look like.
Let $R$ be a unital ring (not necessarily ...

**2**

votes

**1**answer

171 views

### Where does the algebraic closure enter into Block's Theorem?

When applying Block's Theorem on the structure of differentiably simple rings to Lie algebras most authors require an algebraically closed field, but I can see no reference to algebraic closure in ...

**5**

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**0**answers

208 views

### A roadmap to learn about finite-dimensional commutative associative real or complex unital algebras

I've always been secretly fascinated with the rich structure and applications of finite-dimensional associative unital algebras over complete fields. In particular, I am very much interested in the ...

**1**

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**0**answers

103 views

### A simple Lie algebra with modules of a particular type

I’m trying to copy the following construction of P. Forster in group theory for Lie algebras. He takes a non-abelian simple group E which has an FpE-module V such that R = Rad(V ) is faithful and ...

**3**

votes

**1**answer

139 views

### Is every $n$-ary semigroup a subalgebra of an algebra derived from a binary semigroup?

Let $(A,f)$ be an $n$-ary semigroup ($n \ge 2$). Then there exists a ($2$-ary) semigroup $(\overline A,*)$ with an inclusion homomorphism $A \hookrightarrow \overline A$ such that that the ...

**1**

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**1**answer

77 views

### how to write down comatrix of the exceptional Jordan algebra

Suppose we have exceptional Jordan algebra, which is a $3\times3$ matrix ...

**1**

vote

**1**answer

137 views

### A characterization for the ideals of $A+XB[X]$ and $A+XB[[X]]$

Let $A \subseteq B$ be an extension of commutative rings with identity. Then $A+XB[X]$ and $A+XB[[X]]$ are the polynomial and power series rings over $B$ whose constant terms are in $A$. Is there any ...

**3**

votes

**1**answer

313 views

### Maximal ideal of group ring

Let $R$ be a finite commutative ring with identity and $G$ an finite abelian group. Is there any more conditions (on $R$ or on $G $) under which we can characterize maximal ideals of group ring $RG $, ...

**2**

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**0**answers

94 views

### What is the relation (if any) between Clifford algebras and Azumaya algebras?

Suppose the base field is $\mathbb{C}$ and the Clifford algebra is the classical one (i.e. associated to a quadratic form in $n$ variables). It seems that there are relations between Clifford algebras ...

**7**

votes

**1**answer

247 views

### Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra

Definitions
Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...

**3**

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**0**answers

397 views

### Some counter examples in group theory

In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in ...

**5**

votes

**2**answers

126 views

### Positive Elements of a $\ast$-Algebra

In a $C^*$-algebra ${\cal A}$, a positive element is a one of the form $aa^*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a cone, and that for $a,b$ two non-zero ...

**7**

votes

**1**answer

221 views

### Fuzzy logic of Godel

In Gödel logic, is conjunction definable from implication, negation , and disjunction?
We know that conjunction in that logic is not definable from negation and implication.

**5**

votes

**1**answer

479 views

### vector spaces with uncountable dimension and a nice basis

Unlike the reals considered as a vector space over the rationals, I know of a number of nice examples of vector spaces with uncountable dimension that have a nice basis.
For example, the space of ...

**2**

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**0**answers

69 views

### Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons"
though certainly there are others.
Consider the following ...

**0**

votes

**2**answers

143 views

### Making idempotent element by a relation [closed]

Let $R$ be a commutative ring with identity and let $a, b \in R$ such that $a=ab$. How can we make a non zero idempotent element of $R$ by this relation?