1
vote
2answers
37 views

Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
1
vote
0answers
25 views

Composition series of finite length modules

Given a field $K$ of characteristic zero. Let $V$ be a finite dimensional $K$-vector space and let ${f} \in End_K(V)$. Then $V$ can be regarded as a $K[X]$-module by the action: ...
2
votes
0answers
79 views

Help in this proof in Lang's Algebra book (really elementary doubt)

My doubt is really elementary: Let $(f_1,\ldots,f_n)$ be a row matrix with some component with leading coefficient $1$ and $d$ the smallest degree of a component of $f$ with leading coefficient $1$. ...
0
votes
0answers
20 views

Differential operators in the language of modules

I am reading articles about differential operators. Authors try to treat differential systems , by studying the differential operator on a vector space in the language of modules. In this manner ...
2
votes
1answer
27 views

Tensor product of certain algebras

If S is an $R$-algebra of finite dimension, and $A$ is an algebra of infinite dimension, then is $S \otimes S \otimes A \cong S \otimes A$?
4
votes
0answers
48 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
0
votes
0answers
19 views

Linear Differential Equation with Quasiperiodic Coefficient: A Question

I am reading the book Synergetics. The following passage is from the book: Assume that in the differential equation $\dot{q}= a(t)q$, where $a(t)$ is quasiperiodic, i. e., we assume that $a(t)$ ...
4
votes
1answer
90 views

Artin exercise on free modules homomorphism

2.3 Let $A$ be the matrix of a homomorphism $\varphi:\mathbb Z^n\to\mathbb Z^m$ of free $\mathbb Z$-modules. (a) Prove that $\varphi$ is injective if and only if the rank of $A$, as a real ...
1
vote
2answers
52 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...
1
vote
1answer
28 views

direct products and direct sums of skew fields

If F is a skew field then are the arbitrary direct sums of F-modules isomorphic to their direct products (over the same index). I mean, if R is a division ring, and $\{M_i\}_{i\in I}$ some familly ...
2
votes
1answer
43 views

Computing the kernel of a homomorphism from a free $\mathbb{Z}$-module

Given a finitely generated free $\mathbb{Z}$-module $N$ and a homomorphism $\varphi:N\to M$, is there an easy way to compute the kernel of $\varphi$? Since the kernel is also a free ...
2
votes
1answer
62 views

Can we classify commuting pairs of matrices up to conjugacy?

Recall that two $n\times n$ matrices over $\mathbb{C}$ are conjugate if and only if they have the same Jordan canonical form. Question. Is there a similar classification for commuting pairs of ...
3
votes
1answer
50 views

Determinant of a Linear Transformation is Unique Up to a Unit

Can anyone explain the following assertion from a textbook I'm reading? "...$R$ is a Dedekind domain, $K$ its quotient field, $U$ a finite dimensional vector space over $K$ of dimension $n > 0$ ...
5
votes
1answer
153 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
1answer
32 views

Duals of modules over algebras

Suppose $A$ is an associative and commutative $\mathbb{R}$-algebra with unit $1_A$ and $M$ is an $A$-module. Let's write $\cdot_A:A\times M\to M$ for the $A$-scalar multiplication on $M$. 1.) It ...
0
votes
2answers
124 views

What's a Dual Basis?

If $L/K$ is a finite extension of fields with $v_1, ... , v_n$ a basis, then we have an isomorphism of $K$-modules $L/K \rightarrow Hom_K(L,K)$, where a basis for $Hom_K(L,K)$ is the "dual basis" of ...
1
vote
1answer
61 views

Invariant Factor Decomposition of Integer Matrices

Suppose $M$ is an $n \times n$ matrix with integer entries. If we think of $M$ as acting on $V=\mathbb{Q}^n$, then we can think of $V$ as a $\mathbb{Q}[M]$-module and have that $V$ is isomorphic to a ...
0
votes
0answers
16 views

System of linear equations with modulo elements

Transform the following $3 \times 3$ matrix $A$ with entries in $\mathbb{Z}/5\mathbb{Z}$ into reduced row echelon form (with $[1]$ as Pivot element and $[0]$ above the Pivot element). State out the ...
0
votes
1answer
36 views

field extension $F(x)=F(x^2)$ [duplicate]

Let $x$ be algebraic over $F$ such that the field extension $F(x):F$ satisfies $[F(x):F]$ odd. Then prove $[F(x):F]=[F(x^2):F]$ hence $F(x)=F(x^2)$. How to prove? I only obtained the proof for the ...
4
votes
3answers
129 views

similar matrices over $\mathbb{Z}$

Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
0
votes
1answer
70 views

matrices with same minimal and characteristic polynomials are conjugate

Let $A$, $B$ be two $n\times n$ matrices over $\mathbb{C}$. If $A$, $B$ have the same minimal polynomials and characteristic polynomials, then can we prove that $A$, $B$ are conjugate? i.e., does ...
1
vote
0answers
36 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 ...
3
votes
1answer
64 views

Proof of Clifford's theorem for modules

http://en.wikipedia.org/wiki/Clifford_theory#Proof_of_Clifford.27s_theorem I've a very easy question that I just can't seem to find the answer to. I'm self-studying so I can't ask anyone else. ...
1
vote
1answer
48 views

Any submodule U of V such that the module V/U is completely reducible must contain the radical?

Want to prove: Given an $R$-Module $V$ and $rad(V)$ which I define to be the intersection of all maximal submodules of $V$. I want to show that if, for some submodule $U$ of $V$, we have $V/U$ ...
1
vote
1answer
16 views

Tensor over division rings

Let $D$ be a division ring containing a field $k$ in its center such that $D$ is finite dimensional over $k$. Consider the left $D$-module $D \otimes_{k} V$ where $V$ is a finite dimensional k-vector ...
0
votes
1answer
42 views

Prove equivalence: Internal Direct Sum of Modules (HW)

Let $M$ be a module and $M_1, M_2, M_3,..., M_n$ be submodules of $M$ verify that the following are equivalent: (1) $M$ is the internal direct sum of the $M_i$'s. (i.e. ...
1
vote
1answer
61 views

Homework basic abstract algebra

My question is as follows: $R$ is a ring such that for all $x \in R, x^2=x$ $p$ is a prime ideal of $R$. Show that $R/p$ (R modulu p) has exactly 2 elements. What I did: $x^2=x$ $x^2-x=0$ ...
1
vote
0answers
68 views

Irreducible modules - semisimple algebras and endomorphism rings

Let $A$ be a finite dimensional, semi-simple $k$-algebra and $V$ and irreducible $A$-module. I am trying to prove the following claim: If $B = \text{End}_A(V^{\oplus r})$ then $$W = ...
3
votes
0answers
104 views

R-modules over a vector space

Let $V = \mathbb{C}^3$, $A$ be the matrix with column vectors $e3, e1, e2$ (where $e1, e2, e3$ are the standard basis vectors for $\mathbb{R}^3$). $R = \mathbb{F}[X]$. Let $V_a = V$ be the R module ...
1
vote
0answers
166 views

Spinor representation and Clifford modules

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$. For a finite-dimensional $\mathbb Z_2$-graded complex ...
0
votes
1answer
46 views

There's a bijective correspondence between complements of a submodule $N \leqslant M$ and extensions of $\sigma : N \approx M_1$.

Let $N$ be a submodule of $R$-module $M$. let $\sigma : N \approx M_1$ be an $R$-isomorphism. Then the correspondence $H \to \bar{\sigma}$, where $\ker(\bar{\sigma}) = H$ is a bijection from the ...
1
vote
0answers
54 views

A basis of a module is a minimal spanning set.

This is from Roman's Advanced Linear Algebra. Theorem 4.4 Let $\mathcal{B}$ be a basis for an $R$-module $M$. Then $(1) \ \mathcal{B}$ is a minimal spanning set. $(2) \ \mathcal{B}$ is a ...
6
votes
1answer
214 views

Obtain a basis of invertible matrices for $M_n(D)$, where $D$ is an integral domain

Let $D$ be an integral domain. Prove that $M_n(D)$ has a basis consisting of $n^2$ invertible matrices. Consider $M_n(D)$ as a $D$-module. Define invertible elements $V_{nn}=I_n$, $V_{n-1,n-1} = ...
3
votes
1answer
117 views

Similar matrices

Let $A$ be a complex $n×n$ matrix with $tr(A) = 0$. Show that there exists an invertible $n×n$ matrix $B$ such that all diagonal entries of $BAB^{-1}$ are zeros. $\bf Edit:$ Is the fact that $\forall ...
1
vote
1answer
26 views

Is the dual of an equivariant metric equivariant?

Let $g$ a finite dimensional $K$-vector space, and let $g:V \otimes V \to K$ be an inner-product. If As usual, we can use the musical isomorphisms of $g$ to define an inner product on $V^*$, which we ...
3
votes
2answers
79 views

Example of modules over PID's

From Advanced Modern Algebra (Rotman): Definition If $M$ is a finitely generated torsion $R$-module, where $R$ is a PID, and $$M= R/(c_1) \oplus R/(c_2) \oplus ... \oplus R/(c_t),$$ where $t \geq ...
0
votes
1answer
130 views

Infinite tensor product definition

My question is short and popular: how to define an infinite tensor product of modules over a ring? So, there is an infinite set $I$ and $A$-modules $M_i$. I should understand what $\otimes_{i\in I} ...
5
votes
2answers
303 views

How does one obtain the Jordan normal form of a matrix $A$ by studying $XI-A$?

In our lecture notes, there's the following example problem. Find a Jordan normal form matrix that is similar to the following. $$A=\begin{bmatrix}2 & 0 & 0 & 0\\-1 & 1 & 0 ...
3
votes
1answer
68 views

Hamiltonian Quaternions: A Call for Counterexample for ($AB=I_m \implies n≥m$)

How can I build a counterexample to $AB=I_m \implies n≥m$ in the ring of Hamiltonian quaternions? Notice that the vectors $(1,i)$ and $(j,k)=(1,i)j$ are linearly dependent as vectors of the right ...
4
votes
1answer
242 views

Prove $(2, x)$ is not a free $R$-module.

Let $R = \mathbb Z[x]$ and let $M = (2, x)$ be the ideal generated by $2$ and $x$, considered as an $R$-module. Show that $\{2, x\}$ is not a basis for $M$. Show that any 2 elements of $M$ are ...
0
votes
0answers
72 views

What does “singly generated” mean?

Like the title says, what's the meaning of "singly generated", like in "singly generated Hilbert module". I never came across such terminology and I can't find a definition.
1
vote
1answer
49 views

Image of the projection map onto an irreducible module

Let $A$ be a $K$-algebra $V$ be an $A$-module.Let $V=\bigoplus W_j$ with $W_j$ irreducible for all j .Let $M$ be an irreducible $A$-module. And $N=\Sigma \{W_i: W_i$ is isomorphic to $M\}$. Now let ...
1
vote
1answer
137 views

Is there a unique cyclic submodule for every factor dividing the order of a cyclic module?

Suppose $\langle v\rangle$ is a cyclic module over a PID of order $a$. Then there is a cyclic submodule of order $b$ for any $b$ dividing $a$. Namely, $\langle cv\rangle$, where $c$ is such that ...
2
votes
2answers
59 views

What is an easier way to think about tensor products of modules?

I'm having trouble understanding the the tensor product of modules. Sure I get the "it's quotient of the free abelian group by the subgroup generated by certain elements" construction, but this ...
2
votes
1answer
203 views

System of Linear Equations using Mod

I just want to check that I did a certain problem correctly. This is it: $$a+b=3 \pmod{26}\\2a+b=7 \pmod{26}$$ Solve for $a$ and $b$ Now I setup the augmented matrix: $$\left[ \begin{array}{ccc} 1 ...
1
vote
0answers
163 views

ISBN Code - Construction, check digit example

I am doing a homework example and would like to ask your opinion on the correctness of the example: How is the ISBN 3-519-42227-1 constructed, at what point is the check digit and how to calculate ...
1
vote
1answer
47 views

About the basis of a module

Let $n\in \mathbb Z, n\ne 0$. Prove that $ \mathbb Z/n\mathbb Z$-module $\mathbb Z/n\mathbb Z$ has a basis, but $\mathbb Z$-module $\mathbb Z/n\mathbb Z$ hasn't any basis. Hope everyone help me with ...
7
votes
3answers
198 views

The connection between decomposition of $\ker(f_\phi g_\phi)$ and the Chinese remainder theorem

Original problem Suppose $K$ is a field, $f,g\in K[x]$ and $\gcd(f,g)=1$. $V$ is a linear space based on $K$ and $\phi\in\operatorname{End}(V)$. Notation $f_\phi$ and $g_\phi$ denote linear ...
0
votes
2answers
87 views

Suppose R is a commutative ring, $A \in R_n$, and the homomorphism $f: R^n \to R^n$ defined by $f(b) = Ab$ is surjective. Show $f$ is an isomorphism.

How would I go about showing this? Suppose $R$ is a commutative ring, $A \in R_n$, and the homomorphism $f: R^n \to R^n$ defined by $f(b) = Ab$ is surjective. Show $f$ is an isomorphism. (Edit: ...
18
votes
3answers
297 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 ...