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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.

In abstract algebra, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. On the other hand, free abelian groups are precisely the free modules over the ring $\Bbb Z$.

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Existence of maximal free submodules

Let $A$ be a domain and $M$ be a finitely generated module. Is there a free submodule which is maximal among free submodules? Answer is yes for Noetherian rings, obviously. Also the rank of any such submodule is determined by $M$ by tensoring up…
Davide Pierrat
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How to prove a specific quotient of polynomial ring is a free module?

Problem 1.19 in Eisenbud's Commutative Algebra asks the following. Given $R = k[x,y,z,w]$ and $I = (yw - z^2, xw - yz, xz - y^2)$, show that $R/I$ is free as an $k[x,w]$-module, and exhibit a basis. It is simple to show that under the desired…
C.D.
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Does "every submodule of a free module under a PID is free" imply the Axiom of choice?

It is a fact that under the Axiom of choice, every submodule of a free module under a PID is free. A proof is available at https://math.stackexchange.com/a/162958/1389108. My question is, does the implication go the other way? Suppose I have the…
Dennis Chen
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Is there any relation of injective modules to free modules?

Projective modules are direct summands of free modules. As I perceive it, projections and injections are dual notions. Based on that, I was looking whether there is a relation of injective modules to free modules (similar to the natural relation of…
Manos
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A surjective homomorphism between finite free modules of the same rank

I know a proof of the following theorem using determinants. For some reason, I'd like to know a proof without using them. Theorem Let $A$ be a commutative ring. Let $E$ and $F$ be finite free modules of the same rank over $A$. Let $f:E → F$…
Makoto Kato
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Generators of a finitely generated free module over a commutative ring

Let $L$ be a finitely generated free module over a commutative ring $A$. Set $n=\operatorname{rank} L$. Let $x_1,\dots,x_m$ be generators of $L$. Then $m \ge n$? If $m = n$, then is $x_1,\dots,x_m$ a basis of $L$?
Makoto Kato
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Tensor product of free modules

Suppose $M$ and $N$ are free $R$-module($R$ is a commutative ring). The tensor product of $M\otimes_R N$ is free $R$-module? I know for projective modules it is true. How should we build its basis?
Bill
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Are free submodules of finitely generated modules finitely generated?

Are free submodules of finitely generated modules finitely generated? It feels like this should be true because it seems weird that a finitely generated module will have an infinite linearly independent subset, but I am unable to prove it. Assume…
bedune4
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Step in Bourbaki Commutative algebra exercise

I'm trying to solve this step from a Bourbaki's commutative algebra exercise (Chapter I, Flat Modules, exercise 23 (a) page 47, implication $(\gamma)\Rightarrow(\delta)$): Let $R$ be a ring, $F$ be a free $R$-module and $K\subseteq F$ be an…
Fabio Lucchini
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Do free modules over ring without identity exist?

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? Context Edit (by jgon): The original author appears to have abandoned the question, as there are no new comment replies, and…
tyc31316
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Free vector space over a set

Given a set $S$ and a field $F$ we can construct the $F$-free vector space over $S$ in the following way. Consider the set of formal sums $$FS:=\left\{\sum_{s\in S} \alpha_s s\,:\, \alpha_s=0\, \text{except for a finite number of}\, s \in…
eleguitar
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A morphism of free modules that is an isomorphism upon dualization.

Let $R$ be a commutative ring and let $M,N$ be free modules over $R$ and suppose we have a map $f: M \rightarrow N$ such that upon taking $\text{ Hom}(-,R)$ we get an isomorphism $f^* : N^* \rightarrow M^*.$ Must $f$ be an isomorphism as well? …
Twistediso
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Can you lift an automorphism of a finitely generated module to an automorphism of a finitely generated free module?

Let $R$ be a commutative Noetherian ring, and let $M$ be a finitely generated $R$-module with a fixed automorphism $f$. Can one lift $f$ to an automorphism of a finitely generated free module $R^n$? In other words, is there an automorphism $\tilde…
Andrea B.
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When is every direct product of a ring also a free module?

Let $R$ be a non-zero commutative (unital) ring, such that the direct product $R^X$ is a free $R$-module, for any set $X$. What can be said on $R$ ? For instance, does it have to be a field / artinian / local … ? What happens if we assume that $R$…
Watson
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If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring?

Suppose I have a graded polynomial ring $k[x_1,\ldots,x_n]$ on homogeneous generators, where $k$ is a field and the $x_i$ indeterminates, and further that I have a homogeneous graded subring $A$ such that $k[x_1,\ldots,x_n]$ is made a free…
jdc
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