**0**

votes

**0**answers

32 views

### A constrained positive polynomial

Is there an example of a polynomial $Q(x)\in\Bbb Z_{\geq0}[x]$ with $Q(0)=1$ so that $Q(x)=Q_m(x)Q_+(x)$ where $$Q_+(x)\in\Bbb Z_{\geq0}[x]$$ $$Q_m(x)\in\Bbb Z[x]\mbox{ so that }Q_m(x)\mbox{ has at ...

**0**

votes

**0**answers

295 views

### Looking for product of symmetric polynomials evaluated at roots of unity

Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + ...

**2**

votes

**1**answer

145 views

### When does $\overline{U(0,1)}=B(0,1)$ hold?

Given $R$ an absolute valued ring (with unit), sometimes $\overline{U(0,1)}=B(0,1)$ (for example, $\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{H}$) and sometimes $\overline{U(0,1)}\neq B(0,1)$ (for ...

**18**

votes

**6**answers

3k views

### Expressing adj(A) as a polynomial in A?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R $ be the coefficients of the characteristic polynomial of $A$: $\mathop{\mathrm{det}}(A-xI) = p_0 + p_1x + \dots + p_n ...

**1**

vote

**0**answers

37 views

### Locally free sheaves of algebras vs. algebra bundles

It is well known that there is a bijective correspondence between locally free sheaves of modules and vector bundles (cf. ...

**4**

votes

**2**answers

266 views

### Relation between Associative algebra and group algebra

Let $A$ be an associative algebra over a filed $k$.
Q) What are the condition we can impose on $A$ such that there exists a $G$ such that $A=k[G]$, the group algebra generated by $G$?
I am ...

**1**

vote

**1**answer

109 views

### Extension of scalars and projective limits

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...

**1**

vote

**2**answers

134 views

### Degree of sum of integral elements over a UFD

Is it possible to generalize Degree of sum of algebraic numbers (especially Pete L. Clark's answer, based on Keith Conrad's answer)
in the following way:
Let $D$ be a (noetherian) UFD of zero ...

**1**

vote

**1**answer

159 views

### a problem about ideals of polynomial rings

Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions
$$
f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i.
$$
Let ...

**0**

votes

**0**answers

23 views

### Extending scalars in Ore extensions

Given an Ore extension $k[y : \delta]$, where the field is not necessarily central is it possible to extend scalars in the usual way, that is, by tensoring with an extension of $k$?
Is $K \otimes_k ...

**7**

votes

**5**answers

736 views

### Noncommutative Localization of a Ring : Complete Construction

I've been looking for the following construction in the literature, but I've only been able to find (very) partial proofs or proofs of special cases.
Let $R$ be a non-commutative ring and $S$ a ...

**5**

votes

**2**answers

408 views

### The number of ideals in a ring

Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.
Let $R$ be a finite commutative ring with identity. Under what conditions the number ...

**6**

votes

**2**answers

250 views

### Strategies for proving a category is Noetherian?

Let $C$ be a small linear category over a commutative ring $R$. A representation of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation ...

**9**

votes

**0**answers

88 views

### When does Hochschild homology commute with infinite products?

Let $A$ be an associative algebra. Its zeroth Hochschild homology $\mathrm{HH}_0(A)$ is the cokernel of the linear map $A^{\wedge 2} \to A$, $a \wedge b \mapsto ab - ba$. I.e. you quotient the ...

**0**

votes

**1**answer

54 views

### Is the following ideal zero-dimensional?

Let $\left\{ p_{k}\in\mathbb{R}\left[ x_{1},\ldots,x_{n-1}\right]
:k=1,\ldots,n\right\} $ be a family of linear polynomials such that
$p_{k}\left( 0,\ldots,0\right) =0$. Let ...

**0**

votes

**0**answers

58 views

### Degree of function field extension in several variables (degree of an endomorphism over an AV)

I just want to know which is the best way to calculate the degree of a function field extension like this
$[\mathbb{F}_q(a,b,c):\mathbb{F}_q(x,y,z)]$
where
$x\mapsto f(a,b,c)$
$y\mapsto g(a,b,c)$
...

**11**

votes

**2**answers

793 views

### Which commutative groups are the group of units of some field?

Inspired by a recent question on the multiplicative group of fields. Necessary conditions include that there are at most $n$ solutions to $x^n = 1$ in such a group and that any finite subgroup is ...

**3**

votes

**3**answers

364 views

### Polynomials of low degree that clone polynomials of higher degree

Let $f(x_1,\dots,x_{16})=(x_1+x_2+x_3+x_4)(x_5+x_6+x_7+x_8)(x_9+x_{10}+x_{11}+x_{12})(x_{13}+x_{14}+x_{15}+x_{16})\in\Bbb R[x]$.
Let $\mathcal{Z}$ be the zero set of $f$ in ...

**5**

votes

**1**answer

410 views

### When quotient of a $k$-algebra by any maximal ideal is $k$?

Let $k$ be a valued field.
Is there a special term for a commutative (Banach) $k$-algebra $A$ such that for any maximal ideal $m$ we have $A/m=k$?
Is there an easy to check criterion that would imply ...

**33**

votes

**16**answers

28k views

### Linear Algebra Texts?

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...

**16**

votes

**1**answer

973 views

### Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?

This is a question in two parts.
Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...

**1**

vote

**1**answer

52 views

### ideals of polynomial ring of two variables generated by two elements

Let $f,g$ be two polynomials in $\mathbb{Z}[x,y]$, given by
$$
f(x,y)=x^4-3xy+y^2,$$
$$
g(x,y)=x^5-4xy+3xy^2.$$
Let $I=(f,g)$ be the ideal in $\mathbb{Z}[x,y]$ generated by $f$ and $g$.
Is ...

**6**

votes

**0**answers

139 views

### How to prove that a projective module is not free?

Let $A$ be a noncommutative (perhaps $\ast$-) algebra (over $\mathbb{C}$) and let $M$ be a projective module defined via a projector $P\in M_n(A)$; i.e. $M=P(A^n)$. Furthermore, assume that all ...

**20**

votes

**7**answers

2k views

### Consequences of not requiring ring homomorphisms to be unital?

As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example ...

**-1**

votes

**2**answers

263 views

### Anick resolution [closed]

I would like to know some applications of Anick's resolution in non-commutative algebras.

**0**

votes

**0**answers

58 views

### Action of semidirect products of cyclic groups

Is there anything known about group actions of $C_{p}\rtimes C_{p}^{*}$ on the ring of real polynomials $\mathbb{R}[X_{1},\ldots,X_{n}]$, where $C_{p}$ denotes the cyclic group of order $p$ and $p$ is ...

**1**

vote

**1**answer

270 views

### Gerstenhaber versus Schouten

In terms of formal definitions, is there any distinction between Schouten and Gerstenhaber algebras?

**1**

vote

**0**answers

245 views

### Total degree of a polynomial

Let $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$ be minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$,
$$\mathsf{F(p)}=a, ...

**12**

votes

**2**answers

377 views

### $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.
Denote the ...

**0**

votes

**0**answers

79 views

### Global dimension of graded Lie algebra

The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm ...

**6**

votes

**2**answers

576 views

### How do I apply the Boolean Prime Ideal Theorem?

I have become aware of an amazing phenomenon from a myriad of questions and answers here on MathOverflow: many of the results that I would typically prove using the Axiom of Choice can actually be ...

**2**

votes

**1**answer

76 views

### Locally nilpotent operators of the Weyl algebra

$\newcommand{\ad}{\operatorname{ad}}$As my recent post (here) did not receive any answers yet, I thought I would ask a similar question in which I'm also interested.
Let $A=$ $^{k \langle x,y\rangle ...

**5**

votes

**0**answers

91 views

### Noncommutative group of invertible ideals of a ring

Let $R$ be a noetherian domain and let $\mathcal{O}$ be an $R$-algebra that is finitely generated and projective as an $R$-module. The set of invertible fractional ideals of $\mathcal{O}$ is a group ...

**4**

votes

**0**answers

41 views

### examples of local, nonsemisimple , nonsymmetric hopf algebras

I'm searching for (a class of) examples of Hopf algebras , which have the following properties:
they should be finite dimensional
they should not be semisimple
they should be local
they should ...

**8**

votes

**2**answers

655 views

### “Composition of Morita equivalences” or “Morita equivalence and the Nakayama functor”

This problem occured to me, when trying to find a Morita invariant for finite dimensional algebras.
Suppose $\Lambda$ and $\Gamma$ are two self-injective $k$-algebras ($k$ being a field) which are ...

**3**

votes

**0**answers

77 views

### Nilpotent operator of the Weyl algebra

For a research project I'm currently working on, I came across the following problem:
Let $A=$ $^{k <x,y> }\Big/_{(yx-xy-1)}$ be the Weyl Algebra over a field $k$ of characteristic $p$, where ...

**47**

votes

**2**answers

4k views

### How would you solve this tantalizing Halmos problem?

1-ab invertible => 1-ba invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric series. In ...

**1**

vote

**1**answer

132 views

### Structure of $\text{Aut}_R(R[X])$

Let $R$ be a commutative ring with identity. I'd like to know how to determine the set $\text{Aut}_R(R[X])$ of all $R$-automorphisms of $R[X]$.
I've proved that all $\sigma\in\text{Aut}_R(R[X])$ ...

**3**

votes

**1**answer

145 views

### Tits-Kantor-Koecher construction for Jordan algebra of symmetric bilinear form

Suppose $f$ is a nondegenerate, symmetric bilinear form on a vector space $V$ over a field $F$. Then $J = V + F\cdot 1$ is a unital Jordan algebra (known as the "spin factor" Jordan algebra when $F = ...

**4**

votes

**1**answer

128 views

### If $R$ is generated by idempotents, then $\text{Ann}(R)=0$?

Let $R$ be a ring (not necessarily commutative or unital) that is generated by idempotents. I'd like to know if $\text{Ann}(R)=0$ must hold. Here I use $\text{Ann}(R)$ to denote the set of all ...

**2**

votes

**0**answers

93 views

### Geometry of rings and semi-rings

Basically, I wonder whether a theory similar to geometric group theory has been or could be developed for rings and semirings.
One direction would be the following. Consider $\mathbb{N}$ (with the ...

**19**

votes

**2**answers

842 views

### What do you do if you believe a problem is undecidable?

While the title of this question is subjective, I hope to make what I'm looking for quite concrete. The first, and main question is this: If you believe that a problem you are working on is formally ...

**1**

vote

**0**answers

37 views

### Goldie's Theorem for Semigroups

Goldie's theorem is a theorem in noncommutative ring theory that gives a clear picture of semiprime Noetherian rings (actually a slightly broader class). Let $R$ be a semiprime Noetherian ring. The ...

**6**

votes

**1**answer

187 views

### Injective dimension of cyclic modules

Let $R$ be a non-Noetherian ring. Is its left global dimension ${\rm{lD}}(R)$ equal to $\sup \{ {\rm{id}}(M) \mid M \text{ is a cyclic $R$-module} \}$? Here $\rm{{id}}(M)$ denotes the injective ...

**1**

vote

**1**answer

113 views

### Example of a left perfect ring with finite left global dimension that is not right coherent

I am looking for an example in noncommutative ring theory. Namely, I am looking for a left perfect ring with finite left global dimension that is not right coherent. It seems to me that should be ...

**0**

votes

**0**answers

137 views

### For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?

For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo ...

**8**

votes

**0**answers

153 views

### What's the analogue of a Young symmetrizer in the Brauer algebra?

According to Schur--Weyl duality, the centralizer of $\mathrm{GL}(V)$ acting diagonally on $V^{\otimes N}$ is the group algebra of the symmetric group $\mathbb S_N$. An equivalent formulation is the ...

**0**

votes

**1**answer

80 views

### Scalar restriction and scalar extension

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This yields the two functors $h_*\colon{\sf Mod}(S)\rightarrow{\sf Mod}(R)$ (scalar restriction) and $h^*\colon{\sf ...

**2**

votes

**1**answer

231 views

### Automorphisms of Clifford Algebras

What are the automorphisms of real Clifford algebras $Cl_{n,0}$? Of course, I'm interested in the case where they are not central simple.

**0**

votes

**0**answers

121 views

### Rational group scheme

Suppose $G$ is a group scheme over a field $k$, i.e., $G$ is a functor from the category $\text{Alg}_k$ of unital commutative, associative $k$-algebras to the category of $\text{Groups}$. Suppose that ...