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Questions tagged [free-modules]

Use this tag for questions about free modules and related notions as projective modules or free abelian groups. This tag should be used together with the tags of abstract algebra and modules.

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Basis of a free module under module homomorphisms

Hello people my question is the following: Let R be a principal ideal domain, F be a free R-module with basis {e1,…,eN} and f:F→F a module homomorphism.The R-submodule imf of F is always free since we ...
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38 views

Pullback of locally free sheaves is locally free

Lemma 17.4.3 states that if $f:X \rightarrow Y$ is a morphism of ringed space, $G$ is a locally free $O_Y$-module, then $f^*G$ is a locally free $O_X$ module. Suppose that $G$ is a locally free $O_Y$...
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1answer
18 views

Is a finitely generated torsion-free R-module free over R if R is an integral domain?

I know this is the case if $R$ is a PID, but PID's are special instances of Integral Domains, so I am wondering if there is a counter-example to the case where R is an integral domain. This post ...
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1answer
48 views

Proof explanation: Every module is the quotient of free module

Let $M$ be a left $R-$ module, it is said (here for example) that $M \approx F(M)/\sim$ The proof is apparently $F(M) \stackrel{\pi}\to M$ where $(0,\dots,1_m,\dots0) \stackrel{\pi}\to m$ and appeal ...
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1answer
49 views

Does the category of finite dimensional free modules over a principal ideal domain have all finite colimits?

A paper I'm reading states this fact without proof, and I can't seem to find a proof for it.
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1answer
36 views

The exact sequence of tensor product

Prove that for all free right $R$-module $F$ and for all exact sequences of $R$-modules $$0\to M\xrightarrow{f}N\xrightarrow{g}P\to 0$$ then $$0\to F\otimes_RM\xrightarrow{1\otimes f}F\otimes_RN\...
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1answer
19 views

Unique representation of zero in terms of the basis of a free module

I am currently reading the book on abstract algebra by Dummit and Foote. They define a free module as follows (p.330): An $R$-module $F$ is said to be free on the subset $A$ of $F$ if for every ...
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1answer
25 views

Example of both finitely and infinitely generated free modules which are direct sum of two non free modules

Does there exist non free $R$-modules $F_0,F_1$ such that $F=F_0\oplus F_1$ be a free $R$-module? 1- If yes then for what kind of rings $R$ there exist such $R$ modules? 2- If yes then does it holds ...
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1answer
38 views

Exact sequence construction

Given an $R$-module $M$ arbitrary, show it is always possible to construct an exact sequence of $R$-modules $$0\longrightarrow K \longrightarrow L \longrightarrow M \longrightarrow 0,$$ with $L$ a ...
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What can be said about the module that is the $n$-tuples of a matrix ring?

Let $M$ be the set of all $N \times N$ complex matrices. The $n$-tuples of $M$, i.e. $M^n$, is a free module. What are some of known properties of this module? Can this module be isomorphic to another ...
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1answer
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Let $X$ be any set. Prove there exist a vector space $V$ such that $X$ is a basis for $V$

Let $X$ be any set. Prove there exist a vector space $V$ such that $X$ is a basis for $V$. For example what would be the Vector space $V$ such that $X=\mathbb{N}$ is a basis? I don't really see any ...
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When is a free submodule $N$ of a free module $M$ the set of solutions of a system of linear equations?

Suppose one has a (commutative) ring $R$ and a free $R$-module $M$ of rank $n$. Let $N \subset M$ be a free $R$-submodule of $M$. Of course, in general, it is not true that $N$ is the set of solutions ...
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$R^{\mathbb N}$ as a free $R$-module.

Suppose that $R$ is a commutative ring. I'm wondering if the space $R^{\mathbb N}$ is a free $R$ module. I know how to prove that it is not a free $R$ module in the case of $R = \mathbb Z$. But the ...
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1answer
31 views

Projective modules in long exact sequences

Let $A$ be a commutative ring (with unit), and let $(P_i)_i$ be projective $A$-modules sitting in a long exact sequence of $A$-modules: $$0 \longrightarrow P_1 \stackrel{f_1}{\longrightarrow} P_2\...
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1answer
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Does the category of $(R,S)$-bimodules contain a free object on one generator?

I believe that in the category of $(R,S)$-bimodules where $R$ and $S$ are rings with identity, then ${}_R R \otimes_\mathbb{Z} S_S$ is a free object on one generator in this category. But, if $R, S$ ...
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1answer
26 views

What does this Module notation mean?

I have to prove that if $\{x_i\}_{i\in I}$ is a base of a A-module M, then: $$M=\underset{i\in I}{\oplus}Ax_i=\underset{i\in I}{\oplus}(x_i)$$ Am I right to assume that I have to prove that $\sum_{i\...
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36 views

lattices in the complex plane from 3 elements

I am asked to find when, for some nonzero complex numbers $\alpha,\beta,\gamma$, the set defined by $\{l\alpha+m\beta+n\gamma|l,m,n\in \mathbb{Z}\}$ is a lattice. Assuming it is a lattice, I ...
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1answer
29 views

Passing from $\mathbb{F}_p$ to $\mathbb{Z}$.

I am reading Serre's book on Lie Algebras and Lie Groups. On Lemma 4.3, he states that If $E$ is a finitely generated $\mathbb{Z}$-module and $\dim(E \otimes_{\mathbb{Z}} \mathbb{F}_p)$ over $\...
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Prove that a finitely generated $\mathbb F_p[t]$-module is a free $\mathbb F_p[t]$-module.

Specifically, I am asked to show that if $(G,\omega)$ is a $p$-valued group of finite rank, (meaning that the associated graded group $grG$ is finitely generated as an $\mathbb F_p[t]$-module), then $...
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3answers
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Factor of a product ring can not be free

I have seen the following property in my class note but I don´t know how to prove, could someone help me? If we consider the product ring $R=R_1\times{}R_2$, then $R_1$ can not be a free right $R$-...
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1answer
55 views

Concern about algebraic integer

To be honest, I don't figure out how to attack this problem: Let $\alpha$ an algebraic integer i.e. there is a monic polynomial $f(x) \in \mathbb{Z}[x]$ s.t. $f(\alpha)=0$. Let $R:=\mathbb{Z}...
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0answers
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On the free vector space $F(V)$ over another vector space $V$

I'm reviewing free modules and tensor products as part of a foray back into abstract algebra. I notice that the Wikipedia article on the tensor product has undergone some changes in the past few years,...
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28 views

Homology of the Torus over a Module

The homology groups $H_n(T,\mathbb{Z})$ for the torus $T$ are $\mathbb{Z} \oplus \mathbb{Z}$ for $n=1$, $\mathbb{Z}$ for $n=0,2$. Zero otherwise. Question: Changing the coefficients in $\mathbb{...
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1answer
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Finding a basis of $S \otimes_R N$ with $N$ a free $R$-module

Let $S$ and $R$ be commutative rings with $f: R \to S$ a ring morphism. Suppose that $N$ is free $R$-module with basis $B$. I want to understand why $N \otimes_R S$ is a free $S$-module where the ...
3
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2answers
65 views

Do free modules over ring without identity exist? [closed]

If {$e_i$} is the generating set of a free R-module M, and there is no unity in R, how does, say $e_1$ exist in M anyway?
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1answer
40 views

When $M\simeq Ker \oplus Im$? Where $M$ is a Module

Suppose that $M=R^n$ and $R$ a PID, where $n\geq 1,$ and suppose $N$ is a submodule of $M$. A complement of $N$ in $M$ is a submodule $P$ of $M$ so that $M=N\oplus P$ (internal). If $A\in M_{n\times n}...
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3answers
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Weird Definition about Presentations of Modules

Definition 1.70 Suppose $(a_{ij})=A\in M_{m\times n}(R).$ Let $L$ be a free module of rank $m$ with basis $\{x_1,..,x_m\}.$ Let $$ y_j=\sum^{m}_{i=1}a_{ij}x_i \text{ for } j=1,...,n$$ and let ...
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1answer
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If $R$ is an n-fir, why free $R$-modules of rank at most $n$ have unique rank?

I am studying the Cohn's book "Free ideal Rings and Localization in General Rings", and there is something he takes for granted and I cannot find its reason in any part of the book. The question is: ...
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0answers
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Is $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ free?

I'm trying to solve a question which asks me to determine whether the quotient $\mathbb{Z}$-module $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ is free. I'm then supposed to find some ...
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0answers
44 views

Axes of a free module over a PID (2)

Let $R^n$ be a finitely generated free module of rank $n > 0$ over a principal ideal domain. I am trying to prove that for every non-zero element $a$ of $R^n$ there is a basis such that $a$ ...
3
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1answer
72 views

$\mathbb{Z}$-Module exercise

I am trying to solve the following exercise on basic module theory and I am stuck. Any help would be more than welcome! So let $M\subseteq \mathbb{Z}^3$ the solutions to the following problem: $-3x+...
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1answer
90 views

Axes of a free module over a PID

Let $R^2 = R \times R$ be a free module of rank $2$ over a principal ideal domain. I am trying to prove that for every non-zero element $a$ of $R^2$ there is a basis such that $a$ belongs to one of ...
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0answers
219 views

The reason why FTFGMPID fails with UFD instead of PID

[Theorem] Let $M$ be a finitely generated module over a PID $R$. Then $$M \simeq R^d \oplus R/(r_1) \oplus \cdots \oplus R/(r_n)$$ for some $d \geq 0$ and $r_1| \cdots | r_n$. If we replace the ...
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1answer
56 views

About dual of finitely generated projective module

Let $M$ be a finitely generated projective module, $x \in M$ and $x \neq 0$. Then is it true that there is $g \in M^*$ such that $gx \neq 0$? If yes how to prove it? For vector space dual this ...
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2answers
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Why is the $C^\infty(M)$-module of smooth sections of a vector bundle $E$ not free?

Let $M$ be a second countable smooth manifold. When I learned about differential geometry, a side note was made about how if $E$ is a vector bundle, $\Gamma(E)$ is a $C^\infty(M)$-Module that is not ...
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1answer
39 views

If $V$ is free, show that $f$ is surjective.

Let $R$ be a commutative ring with $1$. Let $V$ and $W$ be $R$-modules. a) Exhibit a canonical $R$-linear map $f: V^* \otimes V \to R $ b) If $V$ is free, show that $f$ is surjective. Now for the ...
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0answers
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How to prove that for $K$ $\mathbb{Z}$-free & $Q$ projective, $K\otimes_{\mathbb{Z}}Q$ is a projective?

Let $K$ be $\mathbb{Z}$-free and $Q$ a projective $\mathbb{Z}G$ module then $K\otimes_{\mathbb{Z}}Q$ is a projective $\mathbb{Z}G$-module. I believe that this follows from the adjoint isomorphism ...
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1answer
14 views

Every projective module is a submodule of a free module?

I've seen this statement on the internet but I could not find a proof. Actually this is true for any module I think. Can a proof be given as follows? Let $M$ be an $R$-module. Take a generating set $...
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Intersection of parameterized cosets in a free-abelian group

Let $L_1,L_2$ be subgroups of $\mathbb{Z}^m$, and let $\mathbf{P}_1,\mathbf{P}_2$ be $r\times m$ integer matrices. Then, it is straightforward to check that the set \begin{equation} S_{\{1,2\}} := \{...
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1answer
104 views

Linear dependence as a binary relation

Elements $a$ and $b$ from an $R$-module $M$ are linearly dependent if there are scalars $x$, $y$ in $R$, $x \neq 0$ or $y \neq 0$, such that $xa = yb$. Let $M$ be a torsion-free module over an ...
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1answer
24 views

Let $\mathbb{C}[X]$ be a module over itself. Given $z \in \mathbb{C}$ define $M_z = \{ f \in \mathbb{C}[X] \ | \ f(z) = 0\}$. Give a basis for $M_z$

Let $\mathbb{C}[X]$ be a module over itself. Given $z \in \mathbb{C}$ define $M_z = \{ f(X) \in \mathbb{C}[X] \ | \ f(z) = 0\}$. $M_z$ is a submodule of $\mathbb{C}[X]$. Give a $\mathbb{C}$-basis for $...
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1answer
75 views

Show that the associated matrix is $PAP^{-1}$

Let $\eta \in \operatorname{End}_{R}(R^n)$ and let $A$ be the associated matrix of $\eta$ with respect to the basis $(e_1, \dots e_n)$. Let $f_i = \sum p_{i,j}e_j$ where the matrix $P=(p_{i,j}) \in ...
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0answers
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Onto Algebra homomorphism between group rings.

I have to determine onto $F$-Algebra map from group algebra $FS_5$ to $M_4(F)$ where $F$ is any finite field of characteristic $2$ and $S_5$ is symmetric group of degree $5$ generated by $a=(1,2,3,4,...
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1answer
68 views

free resolution of $\mathbb{Z}$ as an $\mathbb{Z}[x]/(x^n-1)$-module

As the tittle says, I am having a bad time thinking on the construction of a free resolution of $\mathbb{Z}$ as an $(\mathbb{Z}[x]/(x^n-1))$-module. I know that I should give an exact sequence of the ...
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1answer
75 views

If ideal $I$ of domain $R$ is free $R$-module, then $I$ is principal ideal.

If ideal $I$ of domain $R$ is free $R$-module, how to prove $I$ is principal ideal? Is this right if $R$ is just a commutative ring? My thought: $(1)$ for non-zero ideal $I$ and $J$, let $0\not=a ...
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74 views

$M$ is free $R$-module $\iff$ $M$ has $R$-basis

We will define the free $R$-modules. Definition. Let $R$ be a ring with $1_R$ and $F$ an left $R$-module. We call $F$ free $R$-module, if $$F=\bigoplus_{i\in I} R_i$$ where $R_i:=\langle b_i \...
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1answer
47 views

Alternative definition on free modules.

I 'm trying to compare Free Modules and Free Abelian Groups. We know that, Definition. An abelian group $G$ is called free abelian group with rank $n\in \Bbb N^*$, if $G$ is the direct sum of $n$ ...
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1answer
57 views

Finitely generated free module is projective.

Call a $R$-module projective if every short exact sequence $0 \to A\stackrel{f} \to B\stackrel{g} \to C \to 0$ of $R$-modules splits. Call a short exact sequence as above split, if it admits a ...
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1answer
60 views

Tensor product of modules.

Please give me a hint for this problem. Let $R$ be a ring, and $I$ a right ideal of $R$. Show if $M$ is a left $R$-module, then $$ f \ \colon (R/I) \otimes_{R} M \to M/IM$$ defined by $ f((r+I) \...
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0answers
33 views

Understanding proof of: A submodule of a free module of finite rank is also free of finite rank?

Sorry in advance for the long exposition; it is necessary to include it. I am trying to understand the proof of this result from Dummit & Foote (so in particular, I can't use the result yet): ...