# 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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### Finitness conditions under change of rings?

Let $R \to S$ be a homomorphism of commutative rings, such that $S$ is finitely generated and projective $R$-module. Proof that if $A$ is a finitely presented $S$-module then $A$ is a finitely ...
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### If $R$ is a right hereditary ring, then any submodule of a right projective $R$-module is again projective.

Recall that a ring $R$ is called right hereditary if every right right ideal $I\subset R$ is projective as a right $R$-module. I need to prove that if $M_R$ is projective module and $N\leq M$ is any ...
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### Free objects in the category of all R-modules without linearly independent generating set

We can define a free module $F$ over a commutative ring with identity $R$ as a free object (with universal property) in the category of unitary $R$-modules. This is equivalent to the existence of a ...
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### Equality of two free modules of the same finite rank under strong hypothesis.

So basically the question is the one of the title of the post, but let me show you the context: Let $\mathbb K$ be an algebraic closed field and let $f\in \mathbb K[X,Y]$ be a polynomial such that is ...
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### Finding a basis for a free module

Let q $\in$ $\Bbb Q^3$, such that the elements of q are positive numbers. Let M $\subset \Bbb Z^3$ be defined as $\hspace{3in}$M = $\{ \mathbf n \in \Bbb Z^3 | \mathbf n \cdot \mathbf q \in \Bbb Z \}$,...
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### $M/xM$ free over $R/xR$ implies $M$ is free over $R$ when $R$ Noetherian

The following is from https://stacks.math.columbia.edu/tag/00NS. I'm having some difficulty understanding some steps of the proof. Let $R$ be a Noetherian local ring. Let $x \in \mathfrak m$. Let $M$ ...
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### Under what conditions would $S^{-1}F$ be a free left $S^{-1}R$-module?

So I am thinking of a ring $R$ with multiplicative identity $1_{R}\ne 0_{R}$, $F$ is a free left $R$-module, and every element of $S$ is not a zero divisor, and $0_{R}\in S$. Under these conditions ...
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### Invariant basis number for matrix algebras

Let $\mathbb{k}$ be a field (we can suppose $k=\mathbb{R}$ or $\mathbb{C}$ if necessary). Let $M_n(\mathbb{k})$ denote the ring of matrices with entries in $\mathbb k$. Recall that a ring $R$ has the ...
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### Proof that any element of a free abelian group can be extended to a basis

Let $A$ be a free abelian group of rank $n$, and let $\alpha_1 \in A\setminus \{0\}$ such that $\alpha_1 \not \in kA$ for all $k > 1$. Do there always exist $\alpha_2,\ldots, \alpha_n \in A$ such ...
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### On splitness of epimorphisms?

Let $\alpha: M \to N$ be an epimorphism of left R-modules. Let $Q$ be a left R-module of finite projective dimension d with a morphism $\beta : Q \to N$. We know that if d=0 i.e, $Q$ is projective ...
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### Being direct summand of free module implies having dual basis.

I need to provet equality of two definitions of projective module: being direct summand of free module (or equally: having embedding into free module) and having dual basis. Lets use Wikipedia ...
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