**18**

votes

**2**answers

981 views

### Is every compact topological ring a profinite ring?

There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...

**0**

votes

**0**answers

41 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 ...

**8**

votes

**3**answers

479 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 ...

**9**

votes

**0**answers

83 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 ...

**1**

vote

**0**answers

48 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 ...

**0**

votes

**1**answer

92 views

### fixed points of an affine polynomial automorphism

Let $K:- k[x_1, x_2, \cdots, x_n]$ be the polynomial ring over a field $k$. Let $a_i, b_i \in K$ where $a_i \ne 0$. Consider the automorphism $\alpha$ of $K$ defined by
$x_i \mapsto a_ix_i + b_i$. ...

**4**

votes

**1**answer

114 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
...

**3**

votes

**1**answer

122 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 ...

**0**

votes

**0**answers

32 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 $\alpha$ ...

**2**

votes

**1**answer

183 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 ...

**2**

votes

**1**answer

359 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.

**2**

votes

**0**answers

70 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 ...

**0**

votes

**1**answer

62 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 ...

**5**

votes

**0**answers

536 views

### 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 ...

**1**

vote

**0**answers

65 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 ...

**25**

votes

**0**answers

587 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 ...

**2**

votes

**2**answers

157 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 ...

**33**

votes

**8**answers

3k views

### What makes a theorem *a* “nullstellensatz.”

I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...

**0**

votes

**1**answer

112 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 ...

**2**

votes

**1**answer

98 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 ...

**-1**

votes

**1**answer

123 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 $x \in M$.
...

**0**

votes

**2**answers

61 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
...

**3**

votes

**0**answers

171 views

### 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
...

**6**

votes

**0**answers

196 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: ...

**4**

votes

**4**answers

325 views

### What is an ideal-supporting algebra?

I'm sorry if this question is too elementary, but I asked it at MathStackExchange and it received no responses.
On the Wikipedia page for congruence relation it mentions how for groups and rings, ...

**0**

votes

**0**answers

106 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 $n$ ...

**6**

votes

**2**answers

211 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 ...

**24**

votes

**3**answers

2k views

### What is interesting/useful about big Witt Vectors?

$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing ...

**12**

votes

**2**answers

1k views

### Maximal ideal that annihilates entire ring

Does there exist a ring $R$ with a nonzero maximal ideal $M$ such that $R^2=R$ and $MR = RM = 0$?
Here $R$ is associative but does not have an identity (obviously). It seems a simple enough question ...

**1**

vote

**0**answers

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

**20**

votes

**8**answers

4k views

### Simplest examples of rings that are not isomorphic to their opposites

What are the simplest examples of
rings that are not isomorphic to their
opposite rings? Is there a science to constructing them?
The only simple example known to me:
In Jacobson's Basic Algebra ...

**2**

votes

**1**answer

142 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
...

**0**

votes

**0**answers

19 views

### Vanishing ideal of a finite set of points does no have expected amount of cones in gröbner fan

I am reading the 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 ...

**8**

votes

**2**answers

412 views

### What is an example of a ring with two (or more) multiplicative right-identities?

I know that if a ring has a multiplicative identity, then the multiplicative identity must be unique. Are there simple-to-describe examples of rings with two (or more) multiplicative right-identities?
...

**3**

votes

**1**answer

98 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 ...

**0**

votes

**0**answers

44 views

### references for semi-linear algebra over fields

What is a good reference for semi-linear algebra? In particular semi-linear transformations and their matrix representations.

**3**

votes

**0**answers

271 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 ...

**9**

votes

**1**answer

227 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 ...

**4**

votes

**1**answer

102 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 $c:B\to B\otimes H$ ...

**13**

votes

**1**answer

207 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 ...

**14**

votes

**13**answers

2k views

### Ring with three binary operations

A rather precocious student studying abstract algebra with me asked the following question: Are there interesting rings where there are not just two but three binary operations along with some ...

**1**

vote

**0**answers

89 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**

votes

**0**answers

208 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

**1**answer

256 views

### Counting Roots of Unit

Let $p\left( x\right) =%
%TCIMACRO{\tprod \limits_{k=1}^{m}}%
%BeginExpansion
{\textstyle\prod\limits_{k=1}^{m}}
%EndExpansion
\left( x^{e_{k}}-\omega_{k}^{e_{k}}\right) $ be a polynomial with
...

**3**

votes

**3**answers

135 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 & ...

**2**

votes

**0**answers

77 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 ...

**6**

votes

**4**answers

360 views

### Does $S$ being a free rank-$n$ $R$-algebra imply that $S/R$ is free rank $n-1$?

Suppose we have a (commutative) ring $R$ and an $R$-algebra $S$. Furthermore, suppose that $S\cong R^n$ as $R$-modules, that is, $S$ is free of rank $n$ as an $R$-module. Can we always choose $1$ to ...

**-5**

votes

**1**answer

83 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?

**5**

votes

**1**answer

203 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, ...

**2**

votes

**0**answers

42 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 ...