# 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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### When $R$-Module of Rank $r$ has a submodule isomorphic to $R^r$?

Let $R$ be an integral domain,$M$ is a finitely generated $R$ module. Prove that rank$(M)=r$ iff $M$ has a free submodule $N \equiv R^r$ , such that $M/N$ torsion . If $R$ is a PID then $N$ may be ...
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### Linear independence of a subset of elements in a finitely generated module to the number of generators of the module

Let $M$ be a finitely generated $R$-module. suppose $m_i, ..., m_k$ and $p_i, ..., p_n$ are elements in $M$ so that ${m_i, ..., m_k}$ is linearly independent over ring $R$, and ${p_i, ..., p_n}$ is a ...
1 vote
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### $P$ is projective implies $P℘$ is a free $R℘$-module

Statement: Let $R$ be a commutative Noetherian ring and let $P$ be a finitely generated $R$-module, then $P$ is projective if and only if $P℘$ is a free $R℘$-module for all $℘$ in $Spec(R)$. For the ...
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### When are the integers a free R-module?

The are some $R$ for which the integers are a free $R$-module. (For example, when $R = Z$.) But there are other cases when the integers are not. (E.g. take $R = Z[G]$ with the action induced by the ...
1 vote
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### Finitely generated module equal to free module of rank $n$? Not just the quotient of free module.

(For commutative algebra) I have a Ring defined as the polynomial Ring over Field $K$: $R = K[x]$. I need to show that the finitely generated module $M$ is equal to a free module of rank $n$ for ...
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### Alternative proof for Structure theorem for finitely generated modules over a principal ideal domain

I'm thinking of an alternative proof for Structure theorem for finitely generated modules over a principal ideal domain (also called Fundamental theorem of finitely generated modules over a PID in ...
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### Understanding excision with another approach

I find a really interesting approach on Excision Axiom with subdivision operator and prism operator defined by a formula (without using induction). The material I followed is public and it has only 5 ...
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1 vote
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### Presentation of $(xy)$ as a $\mathbb{Z}[x,y]$-module.

Let $I=(xy)$ be an ideal of the ring $R=\mathbb{Z}[x,y]$. I want to find the presentation of $I$ as an $R$-module. This may be very simple, but it's betraying my intuition on the subject, so I want to ...
1 vote
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### Do module isomorphisms preserve freeness?

Let $f$ be module isomorphism $f:M_1 \rightarrow M_2$, with $M_1$ free. Then, is $M_2$ free too? I tried to come up with a proof: Let $B_1$ be the basis of $M_1$. Then we know that $f(B_1)$ is ...
I just started studying modules and stumbled upon the fact that: the direct sum of a family of modules is the submodule generated by their union: $$\bigoplus_{i\in I}A_{i}=\langle \bigcup_{i\in I}A_{... 1 vote 0 answers 29 views ### Periodicity of the sequence of ideals generared by the entries of the maps in minimal free resolution of modules over complete intersection ring For a finitely generated module M over a Noetherian local ring (R,\mathfrak m), let I_i^R(M) denote the ideal generated by the entries in a matrix representation of \partial_i, where (F_i,\... 0 votes 0 answers 40 views ### Is it possible to find an algorithm which gives coordinates of the basis elements? Suppose R to be a ring with unity, and we have a module homomorphism f: A \to B, where A,B are R-modules. Suppose the image f(A) is free R-module of rank 2 with R-basis \{u,v\}. We ... 1 vote 0 answers 29 views ### Canonical morphism of inverse limits of inverse systems of modules Let A be a commutative ring and I an index set. Let (M_i, f_i)_{I\in I} be an inverse system of A-modules and let N be an A-module. There is a canonical map$$\phi: (\mathrm{lim}_{\...
I am self studying commutative algebra, and I have a question about an exercise I am working on. Let $I$ be generated by the minors of the matrix: \begin{bmatrix} x & y & z\\ y & z & ...