Skip to main content

All Questions

Filter by
Sorted by
Tagged with
0 votes
1 answer
170 views

Isn't every algebraic operad equipped with a trivial weight?

In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem): Let $P$ be a connected weight graded differential ...
groupoid's user avatar
  • 215
7 votes
0 answers
225 views

Decomposing an endomorphism as a tensor product

$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
Pierre's user avatar
  • 2,287
36 votes
3 answers
2k views

Are large powers of polynomials linearly independent?

Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional. Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
Guillaume Aubrun's user avatar
34 votes
8 answers
4k views

Uncountable counterexamples in algebra

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
1 vote
0 answers
60 views

Bounding the length in a module of evaluated skew polynomials

Let $R$ be a finite principal ideal ring, $S$ a Galois extension of $R$ of degree $m$ (so in particular $S$ is a free $R$-module of rank $m$, and we have an $R$-module isomorphism $S^n \cong \...
JBuck's user avatar
  • 223
5 votes
1 answer
210 views

Relation between row space and column space resp. null space and left null space over general rings

Let $R$ be a ring and $M\in\text{Mat}(R,m\times n)$ a matrix for $m,n\in\mathbb{N}$. What results are known about the relation between column space (cs, image) and row space (rs), resp. null space (...
Thomas Preu's user avatar
1 vote
0 answers
37 views

Bounding the length of an R-module of matrices

Loosely related to this: Bounding the length in a module of evaluated skew polynomials Let $C$ be an $\mathbb{F}_q$-vector subspace of $m \times n$ matrices over $\mathbb{F}_q$. Assume WLOG that $m \...
JBuck's user avatar
  • 223
1 vote
2 answers
333 views

Condition for equality of modules generated by columns of matrices

Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
Rahul Sarkar's user avatar
28 votes
6 answers
5k views

Expressing $-\operatorname{adj}(A)$ 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 $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$. I ...
Laurent Lessard's user avatar
6 votes
1 answer
328 views

Algebra with a certain abelian group as the multiplicative group

Let $A$ be an abelian group. Are there an algebra $\mathfrak{X}(A)$ s.t. the multiplication group is isomorphic to A ? i.e. $$ \mathfrak{X}(A)^{\times} \simeq A. $$ For example, for $A=\mathbb{Z}/4\...
M masa's user avatar
  • 479
1 vote
0 answers
145 views

Has anyone studied this possible generalisation of the Singular Value Decomposition to all commutative $*$-rings?

I don't know any abstract algebraists personally, which is why I'm asking this question here. Let $(R,*)$ be a commutative $*$-ring where $*:R \to R$ is an involution. Each conjecture below is stated ...
wlad's user avatar
  • 4,943
0 votes
1 answer
243 views

A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]

For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define $A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...
user521337's user avatar
  • 1,209
12 votes
4 answers
752 views

Additive commutators and trace over a PID

I would like to find an example of principal ideal domain $R$, such that there exists a square matrix $A\in \mathfrak{M}_n(R)$ with zero trace that is not a commutator (i.e. for all $B,C \in \mathfrak{...
Portland's user avatar
  • 2,829
3 votes
1 answer
191 views

Hermitian forms over $K\times K$

Let $V$ be a finitely generated module over the ring $R=K\times K$ where $K$ is a field. We fix the switch involution on the ring $R$. Let $H$ be a hermitian form over $V$. When $V$ is a free module, ...
Anupam Singh's user avatar
5 votes
3 answers
1k views

adjoint of multiplication operator in a commutative algebra

Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO. Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\...
Tom De Medts's user avatar
  • 6,614
6 votes
2 answers
5k views

Periodic matrices

A square matrix $M$ such that $M^{k+1}=M$, for some positive integer $k$, is called a periodic matrix. Can we characterize the periodic matrices in $\mathcal{M}_n(\mathbb{Z})$? If we replace $\mathbb{...
Portland's user avatar
  • 2,829
6 votes
1 answer
725 views

Who defined and who coined "module"?

The title of my Q. says it all: QUESTION:   Who defined and who coined: module? Would it be Emmy Noether? EDIT   In view of @anon's and KConrad's answers, and as it could have been ...
Włodzimierz Holsztyński's user avatar
2 votes
0 answers
60 views

Linearity of a canonical morphism related to scalar extension and coextension

Let $h\colon R\rightarrow S$ be a morphism of commutative ring. Let $M$ and $N$ be $R$-modules. We consider the canonical morphism of $R$-modules $$p\colon{\rm Hom}_R(S\otimes_RM,N)\rightarrow{\rm Hom}...
Fred Rohrer's user avatar
  • 6,700
3 votes
1 answer
2k 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 Mod}(R)\...
Fred Rohrer's user avatar
  • 6,700
1 vote
1 answer
723 views

Determinants of tensors [closed]

Consider a tensor of dimension $[d]\times[d]\times[d]$ which is symmetric with respect to every permutation of the indices. Are there any $\textbf{explicit}$ formulas for notions like determinant-like ...
Sankeerth's user avatar
2 votes
1 answer
299 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}{\|...
Ma Na's user avatar
  • 309
7 votes
3 answers
2k views

Is there a field which is the union of finitely many proper subfields?

Is there a field which is the union of finitely many proper subfields?
heiko's user avatar
  • 79
3 votes
1 answer
932 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 ...
Fred Rohrer's user avatar
  • 6,700
1 vote
0 answers
95 views

Is it true that the generator of maximal ideal in $M_n(P[x])$ can be choosen to be monic?

Let $P$ be a finite field and $R=M_n(P[x])$ be a matrix polynomial ring. I want to prove that for every polynomial (not necessary with invertible leading term) $A(x)\in R$ such that $R\cdot A(x)$ is ...
Mikhail Goltvanitsa's user avatar
0 votes
1 answer
606 views

Number of Minimal left ideals in the full matrix ring over a finite commutative local ring

Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?
user avatar
1 vote
1 answer
253 views

Chain of ideals in a complex algebra

Suppose $\mathfrak{A}$ is an unital algebra over complex numbers and $\mathfrak{J}$ is chain of left-ideals in $\mathfrak{A}$ ordered by inclusion such that none of its elements is countably generated....
GiroCont's user avatar
1 vote
1 answer
172 views

Does this solution guarantee $det(A)=0$ where $A\in M(R)$? [closed]

Suppose $R$ is a commutative ring with identity $1$ and the following matrix equation holds: $\begin{pmatrix} a_n & & \\ \vdots & \ddots & \\ a_1 &...
booksee's user avatar
  • 398
0 votes
1 answer
170 views

Tensoring with descending chain of modules

Let $A \to B$ be a ring homomorphism. Let $M_1 \supseteq M_2\supseteq \ldots$ be an infinite chain of $A$-modules ($M_i$ not necessarily finite free). Suppose that the limit $\cap_{i=1}^{\infty} M_i$ ...
ringq's user avatar
  • 11
3 votes
2 answers
344 views

Pseudo-idempotent matrix generating a free module

Let $R$ be a commutative ring with $1$. Let $n$ and $k$ be nonnegative integers, and let $A\in\mathrm{M}_n\left(R\right)$ be a matrix such that $A\cdot R^n\cong R^k$ as $R$-modules. Assume that $A^2=\...
darij grinberg's user avatar
3 votes
0 answers
474 views

Jacobson-Bourbaki correspondence

The Jacobson-Bourbaki correspondence induces the traditional, finite Galois correspondence by suitable restriction; I've been pondering two things: 1. Are there any (other) interesting applications of ...
Stephan F. Kroneck's user avatar
3 votes
0 answers
473 views

Infinite Galois correspondence "according to Artin"

Ever since Artin's lectures on Galois Theory one knows how to set up and derive the usual Galois correspondence in the finite(-dimensional) case using just a bit of elementary Linear Algebra, and ...
Stephan F. Kroneck's user avatar