# 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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Let {$A_i$} be ideals of R The ring R can e written as a direct sum $R = {\bigoplus}_i A_i$ if and only if the exists a set $e_1, e_2, ...$ of orthogonal idempotent elements of R such that $A_j = R ... 1 vote 1 answer 113 views ### Question related to "example of free module with bases of different cardinality" In this Planet Math article, it shows an example where a free module has bases with different cardinality. I am having some problems with show why$\phi$is an$R$-module homomorphism. 6 votes 2 answers 81 views ### Question on$(2, 1+\sqrt{-5})$as a submodule of$\mathbb{Z}[\sqrt{-5}]$. Let$R$be the ring$\mathbb{Z}[\sqrt{-5}]$and$I$the ideal generated by$2$and$1+\sqrt{-5}$,$I=(2, 1+\sqrt{-5})$. Show that$I$is not R-module isomorphic to$R$but$I\bigoplus I$is R-module ... 1 vote 1 answer 100 views ### Prove$D$is$R$-algebra I'm studying the paper Galois Theory and Galois Cohomology of Commutative Rings by S. U. Chase, D. K. Harrison and Alex Rosenberg. A couple days ago I posted this question where I wanted to prove that ... 1 vote 0 answers 27 views ### Every left$KG$-module$M$is a right$KG$-module I don't know how to prove that a left$KG$-module is a right$KG$-module. What I have so far is that a left$G$-module is a right$G$-module always defining the operation$x\cdot'g=g^{-1}\cdot x$. But ... 4 votes 3 answers 190 views ### Doubt about the definition of free vector spaces. Introduction: Suppose you wish to construct a set$F(X)$of linear combination of elements of a given a set$X$: $$V = v_{1}x_{1}+\cdots +v_{n}x_{n} \tag{1}$$ where$a\in \mathbb{K}$and$x^{i} \in X$.... 0 votes 1 answer 26 views ### Explanation of Corollary 2.15. Hungerford's Algebra book Corollary 2.15. If$V$and$W$are finite dimensional subspaces of a vector space over a division ring$D$, then$dim V + dim W = dim (V \cap W) + dim (V + W)$. Sketch of proof. Let$X$be a basis of$... 49 views

### 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 ...