1
vote
1answer
41 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
3
votes
2answers
83 views

What kind of algebraic structure is this

I know that a commutative ring with an additional scalar multiplication on it is called an associative algebra. If the ring also has a 1 it is called a unital algebra. What would you call a field with ...
0
votes
1answer
43 views

Ideal of an integral domain all of whose exterior powers are nonzero.

I want to find an integral domain $R$ with ideal $I$ (considered as an $R$-module) such that $\bigwedge^k I\neq 0$ for all nonnegative integers $k$. Dummit and Foote gave the example of $R=\mathbb ...
0
votes
1answer
32 views

If $\alpha$ is $R$-linear epimorphism, $\alpha$ is isomorphism.

I am having a difficult time understanding the proof of a corollary to Cayley-Hamilton theorem in Eisenbud's Commutative Algebra. The statement is: Let $R$ be a ring, and let $M$ be a finitely ...
5
votes
2answers
258 views

Origin of the modern definition of the tensor product

Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
2
votes
1answer
57 views

Nonexistence of a vector space isomorphism

I feel that the $\mathbf{Q}$ vector spaces $\prod_{n=0}^\infty \mathbf{Q}$ and $(\mathbf{Z}-0)^{-1}\prod_{n=0}^\infty\mathbf{Z}$ are not isomorphic, what is the quickest way to demonstrate it? By a ...
2
votes
0answers
29 views

$M\cong N$ iff $[M:N]_R$ is a principal fractional ideal

Let $R$ be a Dedekind ring, $K$ its field of fractions, $U$ a finite vector space over $K$, and $M,N$ finitely generated $R$-modules that span $U$, i.e. contain a basis of $U$. For every $\mathfrak p ...
2
votes
1answer
136 views

Projective equivalence

Definition Two projective plane curves $F$ and $G$ are projectively equivalent if there is a $\varphi_A\in PGL_2(k)$ such that $F(x,y,z)=G(a_{00}x+a_{01}y+a_{02}z,a_{10}x+a_{11}y+a_{12}z, ...
5
votes
1answer
166 views

Generalization of Cayley-Hamilton

I'm having trouble following a proof of this generalization of the Cayley-Hamilton theorem: Suppose that $M$ is an $A$-module generated by $n$ elements, and that $\varphi \in ...
1
vote
0answers
39 views

Vanishing criterion of pure wedges

Let $R$ be a commutative ring, $M$ some $R$-module, and $m,n \in M$. Is there some criterion when $m \wedge n = 0$ in $\Lambda^2(M)$? There are some sufficient criterions, for example that $m \in ...
1
vote
0answers
51 views

Diagonalization and integrality over power/Laurent series rings

The question itself might seem overly specialized and technical (and by this, I mean boring), and it is quite difficult to explain the real motivation for it (but there is!), so I will try to give ...
4
votes
1answer
226 views

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity), then the tensor product $f_1\otimes ...
0
votes
1answer
95 views

Extension of a linear map to a commutative graded algebra

Let's fix the notation, $V=\bigoplus_{i\geq 0}{V^i}$ is a graded vector space and $\Lambda V$ is the free commutative graded algebra on $V$. I have been struggling to understand this example: ...
6
votes
1answer
138 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
1
vote
1answer
68 views

Matrices over a ring: does $PAQ=A'$ imply $\mathrm{Coker}A\cong\mathrm{Coker}A'$?

In A Singular Introduction to Commutative Algebra by Greuel & Pfister, there is written on p. 127: Let $R$ be a commutative unital ring and $A\in R^{n\times k}$, $P\in R^{n\times n}$, $Q\in ...
3
votes
1answer
101 views

Images in a short exact sequence

Suppose $$ 0\to V\to W\to X\to 0\\ \downarrow\quad\quad\downarrow\quad\quad\downarrow\\ 0\to V'\to W'\to X'\to 0\\ $$ is a commutative diagram of vector spaces, with the top and bottom rows short ...
3
votes
1answer
403 views

Can a non-square matrix be called “invertible”?

Let $R$ be a commutative ring with $1 \neq 0$. It is known that $R^{n} \hookrightarrow R^{m}$ implies $n \leq m$ and $R^{n} \twoheadrightarrow R^{m}$ implies $n \geq m$, and I might use this without ...
0
votes
0answers
109 views

Exercise about “dimension of rings”

Let $K$ be a field, and $\mathfrak a\subseteq K[X_{1},\dots,X_{n}]$ the ideal generated by the following polynomials of degree one $$\mathfrak a= \begin{pmatrix} F_{1}=\sum_{i=1}^{n}a_{1i}X_{i} \\ ...
6
votes
2answers
104 views

Can you determine from the minors if the presented module is free?

Motivation (you can ignore this part): A problem in Hartshorne (II.5.8c) asks to show that if we have a coherent sheaf $\mathscr{F}$ on a reduced noetherian scheme $X$, and the function ...
4
votes
1answer
492 views

tensor product and wedge product for direct sum decomposition

If we have a real vector space $V=W_1\oplus W_2$, is it true that $W_1 \otimes W_2 = W_1 \wedge W_2 $? My guess is that this is true. The definition of the $k$-exterior power is the quotient of ...
2
votes
1answer
68 views

Superfluous assumption in a counterexample to Frobenius algebras

In the wikipedia entry on Frobenius algebras, there are some examples and counter-examples. In example 5, where do you need that $\operatorname{char}(k) \neq 2$ ? I think $R:= k[x,y]/ (x,y)^2$ is ...
2
votes
2answers
98 views

generators of an ideal, dimension of a vector space

Let $R$ be a local Noetherian ring (maximal ideal $m$, residue field $k$). Suppose $\{x_{1}, \ldots, x_{n}\}$ generate $m$. Is it true that dim$_{k}(m/m^2) \leq n$?
2
votes
1answer
76 views

Vanishing Ideal of a Linear Subspace

Let $F$ be an infinite field. Let $V$ be a subspace of $F^n$. Let $V^{\perp}$ be the set of all linear functionals $F^n \rightarrow F$ that vanish on $V$. Let $I(V)$ be the vanishing ideal of $V$, ...
7
votes
1answer
208 views

Isomorphism of rings implies isomorphism of vector spaces?

Let $A$ and $B$ be isomorphic unitary rings. Suppose that both of them admit a structure of (maybe finite dimensional) vector space over some field $k$. I would like to know if then $A$ and $B$ are ...
18
votes
3answers
307 views

Bound on nilpotency index of endomorphisms

Let $A$ be a Noetherian ring (commutative with $1$) and $M$ a finitely generated $A$-module. I want to show that there exists a bound $n$ such that for every nilpotent endomorphism $T : M \to M$ we ...
3
votes
1answer
352 views

A non-degenerate trace implies dual basis [updated]

I found a better proof of the theorem in Serge Lang - Algebraic Number Theory but I put in bold the parts I don't understand. Hoping for any explanations of these points. The trace $Tr : L \to K$ ...
1
vote
0answers
89 views

Matrix over a ring with given kernel

Let $R$ be a (commutative, unital,) Noetherian ring and let $M$ be an $m\times n$ matrix over $R$. The columns of $M$ span a submodule $\widetilde M$ of $R^m$, but in general $\widetilde M$ will not ...
3
votes
0answers
363 views

Nullspace of matrix with multivariate polynomial entries

Let $R:=\mathbb{Z}[X_1,X_2,\dots,X_{mn}]$. Suppose $A=(f_{ij})$ is a $m\times n$ matrix with entries in $R$ such that (1)there is no zero column in $A$; (2)for each $i,j$, either $f_{ij}=0$ or ...
2
votes
3answers
205 views

Homomorphism of free modules $A^m\to A^n$

Let's $\varphi:A^m\to A^n$ is a homomorphism of free modules over commutative (associative and without zerodivisors) unital ring $A$. Is it true that $\ker\varphi\subset A^m$ is a free module? Thanks ...
2
votes
0answers
91 views

Is there any other way to prove this statement?

Suppose that $v_1, ..., v_n \in R^m$, where $R^m = M_{m1}(R)$ for $R$ a commutative ring and $R^m$ a vector space. Let the matrix $A= [v_1 |...| v_n]$, the matrix whose $i$th column $= v_i$. I want to ...
3
votes
0answers
87 views

Computation of determinant of a matrix with elements from an arbitrary commutative ring

The cofactor formula for computing the determinant of a matrix is applicable when elements of the matrix are from a commutative ring. However, the complexity of this method is extremely high and I ...
9
votes
1answer
125 views

diagonalizing a matrix over the $\ell$-adics

Let $M$ be a $2 \times 2$ matrix with coefficients in $\mathbb{Z}_{\ell}$ whose characteristical polynomial is $$ P(T) = T^2- (a+d) T + (ad-bc). $$ I've encountered the following assertion: If ...
1
vote
0answers
93 views

Isomorphism induced by the non degenerate form

Let $M$, $N$ be finitely generated projective modules over a ring $R$. Suppose that we have a non degenerate form $\langle\cdot\;,\,\cdot\rangle: N \times N \to R$. ($N$ is isomorphic to its dual ...
10
votes
3answers
261 views

Deducing results in linear algebra from results in commutative algebra

Here are two examples of results which can be deduced from commutative algebra: Any $n\times n$ complex matrix is conjugate to a Jordan canonical matrix (can be proven using the structure theorem ...
5
votes
1answer
169 views

Duality of a finitely generated projective modules

Let $M$ and $N$ be a finitely generated projective module over a ring $R$. Suppose we have a non degenerate bilinear pairing $\langle \ \cdot \ ,\ \cdot\ \rangle: M \times N \to R$. I want to show ...
10
votes
3answers
549 views

Does an injective endomorphism of a finitely-generated free R-module have nonzero determinant?

Alternately, let $M$ be an $n \times n$ matrix with entries in a commutative ring $R$. If $M$ has trivial kernel, is it true that $\det(M) \neq 0$? This math.SE question deals with the case that ...
2
votes
1answer
290 views

Proving $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without the universal property

Let $F$ be a commutative field, and let $U$, $V$, and $W$ be finite dimensional vector spaces over $F$. How can one prove $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without using the ...
1
vote
1answer
366 views

Symmetric power of vector space

Let $V$ be a vector space over a field $k$ of char. zero and denote by $Sym^n_k V$ its $n$-th symmetric power over $k$. Now I simply want to know what $Hom_k(V,Sym^n_k V)$ is for $n \geq 2$. To be ...
5
votes
1answer
155 views

Does similarity of integer matrices imply the transition matrix is an integer matrix?

I'm working on a homework question, and I'm stuck. The question is: Let $A$ and $B$ be $2n \times 2n$ rational matrices with $A^2=B^2=-Id$. The first part of the question asks to show that $A$ and ...
2
votes
2answers
290 views

Is there an analogue of the jordan normal form of an nilpotent linear transform of a polynomial ring?

Is there an analogue of the Jordan Normal Form for an infinite dimensional vector space? In general I think the answer is no. It's been awhile since I studied it, but I'm pretty sure something would ...
3
votes
1answer
134 views

Do a matrix and its transpose have the same invariant factors over a PID?

I suspect this is true since it holds in the case over a field. But suppose $A\in M_{m\times n}(R)$ where $R$ is a PID. Does it still hold that $A$ and $A^{T}$ have the same invariant factors? ...
2
votes
1answer
83 views

Generators for $M_n(\mathbb Q)$

What is the minimum number of generators for $M_n(\mathbb Q)$, the set of $n \times n$ matrices over $\mathbb Q$, which will generate it as an algebra over $\mathbb Q$ ?
1
vote
1answer
53 views

Prescribing linear projection

Let R be a commutative pid, and let M be the free R-module of finite rank k. Given a non-zero proper submodule N of M, does there always exist a projection P such that ker(P)=N? If so, how can we ...
10
votes
1answer
507 views

Tensor products of infinite-dimensional spaces and other objects

It has just occurred to me that most of my intuition for tensor products is derived from the special case of finite-dimensional vector spaces, so I'm wondering which properties I've taken for granted ...
3
votes
1answer
143 views

Finding basis of a free $\oplus_{i=1}^m k$ module in a subspace

I come across this question as I consider a problem dealing with semilocal rings. Suppose that $k$ is a field, $R=\oplus_{i=1}^m k$ is a finite $k$-algebra via the diagonal embedding $k\to R$. Let ...
2
votes
2answers
186 views

Faithful extension(Is the image of a polynomial map an algebraic set? )

Let $k[x_1,\ldots,x_n]$ be a polynomial ring, $k$ be a algebraically closed field. Suppose $k[T_1,\ldots,T_m]$ is finitely generated $k$-subalgebra such that for any proper ideal $I$ of ...