# 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.

216 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $a$ ...
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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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### 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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### 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): ...