# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $P$ ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### Monomial ideals: isomorphism problem for commutative algebras?

Theorem 5.27 in Polytopes, Rings, and K-Theory (Bruns, Gubeladze - 2009 - Springer SMM) claims: Let $K$ be a field and $I\!\unlhd\!K[x]= K[x_1,\ldots,x_n]$ and ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 $1$). ...
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### 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 $$\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)$$ ...
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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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 ...