# Tagged Questions

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votes

**1**answer

345 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 ...

**3**

votes

**0**answers

65 views

### the annihilator of cokernel in a particular case

Let $A\in Mat(m,n;R)$ for $m\le n$ and $R$ a local ring. Consider the $mn\times(m^2+n^2)$ matrix $A\otimes 1_{nn}\oplus 1_{mm}\otimes A^T$, here $1_{mm}$, $1_{nn}$ are identity matrices. I'd like to ...

**13**

votes

**1**answer

384 views

### A weird question about two weird decompositions of $\mathbb{R}$ as a $\mathbb{Q}$-vector space

While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers:
Question: Do there exist a non-trivial ...

**3**

votes

**1**answer

366 views

### Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...

**2**

votes

**0**answers

36 views

### Metabolic vs stably metabolic

Let $A$ be a commutative ring with unit. A non-degenerate symmetric bilinear form $\phi$ on a finitely generated projective $A$-module $P$ is called metabolic if there is a direct summand $L$ of $P$ ...

**4**

votes

**0**answers

153 views

### Dimension of a commuting nilpotent variety

Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...

**2**

votes

**1**answer

28 views

### Separations between notions of rank for modules over commutative (semi-)rings with no zero divisors.

Let $M$ be an $m$-by-$n$ matrix, here are three definitions$^5$ that we could use for rank:
$rk(M) = \min k$ such that for matrices $P$, and $Q$ with $P$ of size $m$-by-$k$ and $Q$ of size ...

**0**

votes

**1**answer

146 views

### Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ...

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**0**answers

121 views

### Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...

**2**

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**0**answers

50 views

### Nullspace of a matrix modulo an ideal

Suppose $R$ is a multivariate polynomial ring and $I$ is an ideal in $R$.
Let $M$ be a $n\times n$ square matrix with entries in $R$, and suppose that det($M$) lies in $I$.
Thus, $M$ has a ...

**0**

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**0**answers

65 views

### Interpret some coefficients in algebras

Let $A$ be a real vector space equipped with a scalar product $\langle \,,\,\rangle$, and assume moreover that a multiplication is define on it so that becomes an algebra (e.g. polynomials with the ...

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votes

**1**answer

214 views

### Characteristic polynomial of exterior power

Suppose $f$ is a linear map, and consider $\Lambda^k f$ as the usual exterior power of $f$ (if you prefer matrices, it is a matrix whose entries are the $k\times k$ minors of $f.$) The coefficients of ...

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votes

**2**answers

276 views

### Variety determined by interior product of the determinant?

Let $\Lambda^k(V)$ be the space of alternating $k$-linear tensors on $V$. Consider the map $f: \left(\mathbb{R}^n\right)^{n-k} \to \Lambda^k(\mathbb{R}^n)$ given by $\left(v_1,v_2, ..., ...

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vote

**3**answers

2k views

### Multiplicative functions $\phi : M_n(F) \longrightarrow F$ with $\phi(I) = 1$

Let $F$ be an infinite field and let $f \in F[x_{11},x_{12},...,x_{nn}]$ be an arbitrary polynomial in $n^2$ variables. Consider the function $\phi : M_n(F)\longrightarrow F$ defined by ...

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votes

**1**answer

109 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 ...

**2**

votes

**3**answers

202 views

### On matrices in linear forms with vanishing determinant

This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...

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vote

**1**answer

171 views

### Classification of pairs of commuting endomorphisms

Let $K$ be an algebraically closed field. I'm interested in isomorphism classes of triples $(V,f,g)$ where $V$ is a finite dimensional $K$-vector space and $f,g$ are commuting endomorphisms of $V$. ...

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votes

**2**answers

131 views

### Dimension of commutative subalgebras of a central simple algebra

let $k$ be a field, and let $A$ be a central simple $k$-algebra over $k$.
What is the maximal dimension of a commutative $k$-subalgebra of $A$?
If $A=M_r(D)$, where $D$ is a central division ...

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votes

**1**answer

348 views

### Maximal size of an almost-disjoint linearly independent family in $K^{\mathbb{N}}$

Let $K$ be a field, say infinite, and denote by $L$ the $K$-vector space $K^{\mathbb{N}}$. What is the maximal cardinality of a $K$-linearly independent subset $X$ of $L$ such that any two distinct ...

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votes

**1**answer

87 views

### Iterated Reduced Tensor Power of Graded Vector spaces

This might be inappropriate for the MO-level. If so I'll delete it...
Suppose $V$ is a $\mathbb{Z}$-graded vector space and
$\overline{T}(V):=V \oplus V\otimes V \oplus \otimes^3 V \ldots$
is the ...

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votes

**2**answers

774 views

### Solve for $A$ and $B$ in $AXB=Y$

Let $R = \mathbb{Z}[x_{1}, \dots, x_{r}]$.
Let $X$ be $n \times n$ matrix with entries in $R$.
Let $Y$ be $m \times m$ matrix with entries in $R$ formed from $\mathbb{Z}$-linear or $\mathbb{R}$-linear ...

**0**

votes

**1**answer

155 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 $\cap_{i=1}^{\infty} M_i$ ...

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**0**answers

244 views

### Algebraic Independence of Polynomials in n Variables with Real Coefficients

I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... ...

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votes

**2**answers

448 views

### Hilbert's Nullstellensatz on polynomials with integer coefficients

Let $f_1, f_2, \ldots, f_m \in \mathbb{Z}[x_1, \ldots, x_n]$. Assume $f_1(X) = f_2(X) = \ldots = f_m(X) = 0$ have no solutions over $\mathbb{C}^n$, then by Hilbert's Nullstellensatz, there exists ...

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**2**answers

1k views

### Over which fields are symmetric matrices diagonalizable ?

The question is motivated by this one real symmetric matrix has real eigenvalues - elementary proof:
Are there other fields $F$ than $\mathbb{R}$ (maybe some valued fields or real closed fields) ...

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votes

**1**answer

157 views

### Dimension of An Ultraproduct Field as a Vector Space over Another Ultraproduct Field

Suppose $F \subseteq K$ are fields with $G$ an ultrafilter on an infinite set $X$. If $F^{\ast}$ and $K^{\ast}$ represent the ultraproducts respectively of $F$ and $K$, it is easy to see that $[K : ...

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**1**answer

221 views

### Linear algebra over principal rings

Consider an extension $R\subseteq S$ of commutative rings, and suppose that $R$ is principal (i.e., $0$ is the only zero-divisor of $R$ and every ideal of $R$ has a generating set of cardinality $1$). ...

**15**

votes

**2**answers

603 views

### a determinantal identity

Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity
$$
\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
...

**2**

votes

**1**answer

84 views

### Dual of a semilinear morphism

Let $R$ be a commutative ring and let $M$ and $N$ be $R$-modules. Let $\sigma:R\rightarrow R$ be a ring automorphism.
Let $f: M\rightarrow N$ be a $\sigma$-semilinear map, i.e. a map of abelian ...

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votes

**1**answer

127 views

### Relations between a set of inner products of vectors

Suppose we have n normalized vectors on an arbitrarily large Hilbert space $|A_1\rangle,\dots,|A_n\rangle$, $\langle A_i|A_i\rangle=1$ for every i. And there're $\frac{n(n-1)}{2}$ inner products ...

**0**

votes

**1**answer

226 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 ...

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234 views

### How many DISTINCT vectors we get from pairs v_i + v_j for some set of given vectors v_i ?

Consider some set of vectors v_i i=1...N , v_i \in Z^k.
e.g. N = 10^4; k = 10
Consider all possible sums: v_i + v_j.
Is it possible to estimate how many DISTINCT vectors we get in advance without ...

**15**

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**1**answer

422 views

### When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?

Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...

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**0**answers

637 views

### Conjugate Matrix

Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix ...

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**2**answers

427 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 ...

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**3**answers

1k views

### Hom(A,C) ⊗ Hom(B,D) injects into Hom(A⊗B,C⊗D): when? why?

Sorry for asking a linear algebra question on a research forum, but this seems to be either a case of extreme blindness on my side, or a case of a result lying much deeper than it seems.
The ...

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**3**answers

392 views

### Multiplicity of eigenvalues in 2-dim families of symmetric matrices

Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...

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**1**answer

298 views

### spurious torsion under compositions of linear maps

Say we have a PID $R$, integers $1 \leq a \leq b$, and $R$-homomorphisms $R^a \stackrel f\to R^b \stackrel g\to R^a$ with $g \circ f$ of full rank.
For $h = f, g, g \circ f$, let $c(h)$ be the ...

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votes

**2**answers

532 views

### Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?

Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed ...

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**0**answers

348 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 ...

**3**

votes

**0**answers

296 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 ...

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votes

**2**answers

268 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 ...

**3**

votes

**4**answers

345 views

### Generalization of Jordan Decomposition for Several Commuting Operators

Recently I became curious about the following question:
Let $V$ be a finite dimensional vector space over $k$ and let $A_1, \cdots, A_n: V \rightarrow V$ be a set of commuting maps. Question: ...

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votes

**6**answers

3k views

### Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find ...

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votes

**3**answers

877 views

### How to define the orientation of a vector space over an arbitrary field?

I know the construction of the Hodge star operator in the context of (pseudo-)euclidean real vector spaces. Apart from the scalar product it involves a orientation of the vector space, which one has ...

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votes

**2**answers

581 views

### Iterated calculation of determinants

Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...

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vote

**0**answers

155 views

### Sum of two free o-submodules in a vector space over a local field

Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...

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votes

**1**answer

1k views

### Commuting Matrices and the Weak Nullstellensatz

In the Wikipedia article on Hilbert's Nullstensatz,
http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz
the following application of the Weak Nullstensatz is mentioned:
Commuting matrices
...

**6**

votes

**1**answer

277 views

### Orbits in commutative groups.

Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization ...

**2**

votes

**2**answers

309 views

### vectors with entries from a finite ring

I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...