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.
525
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Generating set for submodule of free module over a PIR
Let $R$ be a commutative principal ideal ring (not necessarily a domain), $S \cong R^n$ a free $R$-module. We know that any submodule $M$ of $S$ has $m := \operatorname{length}_R(M) \leq n \cdot \...
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47
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Finitely generated $R$-module has a free resolution of length at most $1$?
If $R$ is a PID and $M$ is a finitely generated $R$-module,
How do I show that M has a free resolution of length at most $1$?
So I have to show that M has a free resolution
$....\to F_2 \to F_1 \to ...
- 3,025
1
vote
0
answers
77
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Proving two modules are free based on their direct sum [duplicate]
I have given two modules $M$ and $N$ over a local ring $R$. I also know that $M \oplus N \cong R^n$ for some $n\in \mathbb{N}$. I then have to prove that both $M$ and $N$ are free modules.
Since $M \...
- 139
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38
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Prove that the polynomial ring $R[x]$ is a flat $R$-module.
Here is the question I am trying to solve:
Prove that the polynomial ring $R[x]$ in the indeterminate $x$ over the commutative ring $R$ is a flat $R$-module.
My thoughts:
We know by corollary 42 in D&...
- 3,653
0
votes
2
answers
95
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Why a nonzero finite abelian group is not projective?
Here is the question I am trying to solve (I know it is answered here $A$ be a nonzero finite abelian group then $A$ is not a projective or injective $\Bbb Z$ module. but the answer is not very clear ...
- 3,653
2
votes
0
answers
31
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partial Quillen-Suslin on square matrix
The Quillen-Suslin theorem states that projective modules over $\mathbb{Q}[x_1, \dots , x_n]$ are free. (This also holds over more general fields or rings.)
I have a square $m\times m$ matrix $M$ over ...
- 101
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1
answer
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Construction of the free operad
I am reading Fresse's "Koszul duality of operads and homology of partition posets", and trying to understand the construction of the free operad $F(M)$ on a symmetric sequence $M$ as ...
- 1,653
2
votes
1
answer
59
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Generalized Notion of Krylov Subspaces
Let $\mathcal{X}$ be a vector space over a field $\mathbb{K}$ and let $x_0 \in \mathcal{X} \setminus \{0_{\mathcal{X}}\}$ (here, $0_{\mathcal{X}}$ denotes the zero vector in $\mathcal{X}$). We denote ...
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Are projective modules free over a polynomial ring with infinite indeterminates over a field?
(1) In 1963, Bass proved that any nonfinitely generated projective module is free over a connected Noether ring.
(2) Quillen–Suslin theorem states that any finitely generated projective module over a ...
- 125
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1
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35
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Does this argument use the fact that $\{e_i\}$ is linearly independent at any point?
I'm looking through a proof and I'm trying to tell if it uses a certain fact. If it doesn't, then I think I've figured out a homework problem. Here is the lemma and the proof of it:
Lemma: Every ...
- 967
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0
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52
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How to find an injective ring homomorphism from $S$ to $M_n(R)$,where S is a commutative unitary ring as well as a free module over its subring $R$?
Suppose $S$ is a commutative unitary ring and $R$ is a subring of $S$,naturally S becomes a $R$-module.
To prove: If S is a free $R$-module over R of rank $n$,then there is a ring isomorphism between $...
- 25
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0
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32
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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 ...
- 624
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20
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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 ...
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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 ...
0
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1
answer
47
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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 ...
- 903
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1
answer
107
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Standard decomposition of vector-space endomorphism as direct sum of abelian groups
Let $V=\mathbb Q^7$, $\phi: V \rightarrow V$ the $\mathbb Q$-linear map given by the matrix:
$A=\begin{align*}
\begin{pmatrix} 1&0&0&0&1&0&2 \\0&2&1&0&0&-1&...
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0
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57
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Show that if $Q$ is a non-trivial $\mathbb{Z}$-module, that is divisible, then $Q$ can not be a projective $\mathbb{Z}$-module. [duplicate]
As the question says, I want to show that
If $Q$ is a non-trivial $\mathbb{Z}$-module, that is divisible, then $Q$ can not be a projective $\mathbb{Z}$-module.
Now, here is my reasoning:
Let $F$ be ...
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votes
1
answer
66
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Flatness of $\mathbb{C}[x_1,\ldots,x_n]$ over $\mathbb{C}[f]$, $f \in \mathbb{C}[x_1,\ldots,x_n]$, $n \geq 1$
Let $f \in \mathbb{C}[x_1,\ldots,x_n]$, $n \geq 1$.
Call $f$ 'good' if $\mathbb{C}[x_1,\ldots,x_n]$ is flat over $\mathbb{C}[f]$.
Is it true that every $f \in \mathbb{C}[x_1,\ldots,x_n]$ is good?
$f ...
- 6,415
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0
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42
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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 ...
2
votes
1
answer
55
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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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4
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239
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Every $K[G]$-module is torsionless?
Let $G$ be a finite group and $K$ a field. Consider the group ring $R:=K[G]$. Let $M$ be a (left) $R$-module. Is it true that then there exists a set $S$ and an injective $R$-module homomorphism
$M\...
- 1,653
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votes
1
answer
56
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Free module over a countable set [closed]
Suppose $R$ is a commutative ring. Is $R^{\oplus \mathbb{N}}$ isomorphic to $R^{\oplus \mathbb{N} }\oplus R^{\oplus \mathbb{N}}$ as $R$-modules? If so, how do I find an explicit isomorphism?
Edit: I ...
- 25
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votes
1
answer
31
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Group algebra is a free module of finite rank over any group sub-algebra
Let $G$ be a finite group and let $H$ be a subgroup. Let $\mathbb CG$ and $\mathbb CH$ be the corresponding group algebras. Clearly, the injective group homomorphism $H\to G$
defines a $\mathbb CH$-...
- 912
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vote
1
answer
45
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rank of a free submodule of a free module of infinite rank
I am currently studying the free modules and I am stuck in the following question. Please help me.
I know that if $M$ is a free module on an infinite subset $A$ over a ring $R$ (not necessarily ...
- 400
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0
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16
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Find size of equivalent class of word
I know that the word problem is undecidable in general, but I would like to know about a related but similar problem which is: given a free monoid with a set of rewriting rules, and a string from that ...
- 160
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0
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39
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If $A$ and $C$ are free modules and $C$=$A \oplus B$, need $B$ be free? [duplicate]
I stumbled across this seemingly trivial problem while trying to solve it as a subproblem for an exercise in Weibel's Homological Algebra:
Given a commutative ring $R$ and three $R$-modules $A$, $B$ ...
- 9
1
vote
1
answer
109
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Eisenbud Exercises 1.22 & 1.23 - Understanding Free Resolutions
CONTEXT
The exercises read:
Exercise 1.22: Let $R=K[x]$. Use the structure theorem for finitely generated modules over a principal ideal domain to show that every finitely generated $R$-module has a ...
- 61
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votes
0
answers
10
views
Embedding of a topologically free algebra in another topologically free algebra
Let $k$ be a field of characteristic zero. By definition, a $k[[t_1, ...., t_n]]$-module $M$ is called topologically free if there is an isomorphism of $k[[t_1, ...., t_n]]$-modules $M\cong V[[t_1, ......
- 912
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votes
0
answers
37
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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 ...
- 1,857
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votes
2
answers
56
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Rank and dimension dilemma for modules
Let $R$ be a field of characteristic zero, and $M$ be a $\mathbb{Z}$-module. Knowing that $R\otimes M$ is a vector space, can we deduce that $M$ is a free $\mathbb{Z}$-module and $\text{rank}(M)=\...
- 1,857
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votes
1
answer
73
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How is $\text{Tor}_1^{\mathbb{Z}}(X,R)$ defined in a simple manner as an R-module?
Wishing to understand the universal coefficient formula for homology modules, I came across this new space $\text{Tor}_1^{\mathbb{Z}}$ that I presume that it can be viewed as an $R$-module, and I do ...
- 1,857
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votes
0
answers
39
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Prove a set is a basis for a free module if its elements share coefficients with an actual basis.
Let $R$ be an integral domain, let $R^n$ be a free $R$-module with basis $\{v_i\}_i^n$, let $\phi:R^n \to R^n: v_j \mapsto \sum_i^n m_{ij}v_i$ be an $R$-module homomorphism, let $F$ be $R$-module ...
- 21
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1
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48
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Proving that a subset of a free module is a basis using category theory.
Setup: (see commutative diagram below) Suppose $E = \{\sum_j^n m_{ji}x_j\}_i^n$ is a basis for the free $R$-module $F$. Let $I = \{1, \dots, n\}$ be an indexing set and let $V = \{v_1, \dots, v_n\}$ ...
- 21
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votes
0
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121
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Exact conditions so that vector-space-like change of basis is possible in a free module
This is a follow-up of this question. Suppose $R$ is a unital ring that is not necessarily with the IBN property, and $M$ a nontivial free left $R$-module with a basis $B$.
Under what exact ...
- 949
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votes
1
answer
114
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Change of basis in general free modules
Suppose $R$ is a unital ring that is not necessarily with the IBN property, and $M$ a nontivial free left $R$-module with a basis $B$. It seems to me the following proposition is correct, which makes ...
- 949
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$R[x]/(f)$ is free over $R$ implies that $R[x]/(f) \to Q \otimes R[x]/(f)$ is a monomorphism
I have a question about the proof of corollary 4.12. in Eisenbuds "Commutative Algebra with a view towards Algebraic Geometry":
If $R$ is a normal domain, then any monic irreducible ...
- 777
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votes
1
answer
31
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Finitely-Generated modules over a commutative ring with $1$.
Let $A$ be a commutative ring with $1$. Suppose that every module $M$ over $A$ is finitely.generated with rank $n$ and every submodule $N \leq M$ has rank $r \leq n$. Show $A$ is a principal ideal ...
user1120269
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1
answer
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Does sub-module of free module over PID is free require commutativity?
I read the proof in Lang's Algebra (page 880) that submodules of free $R$-modules for $R$ a PID are themselves free. I can't figure out where the proof requires commutativity of $R$. Is it possible to ...
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48
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Proving an algebra is nilpotent
I recently have started studyng about free algebras and I want to know what kind of methods or approaches are there to prove that free algebra $A$ is nilpotent with nilpotency index $n.$
Let me remind ...
- 19
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0
answers
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Is it known the $S_n$-module structures of free anti-commutative algebra
I recently started studiyng representation theory, especially I am interested in $S_n$-module structures of free algebras over some variety. I know that the $S_n$-module structures are known for some ...
- 19
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0
answers
55
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Comparing Ranks of Free Modules
Let $R$ be a commutative ring and $M$ and $N$ be a free module module over $R$. The the rank(may infinite) of $M$ and $N$ are determined uniquely.
If there is a injection $M$ to $N$ ,then the cardinal ...
- 91
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votes
1
answer
43
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Modules isomorphic to itself direct product.
I'm trying to reach the solution of 1 problem from the book "Modules and Rings: A translation of Moduln und Ringe" of F. Kasch. The problem is:
"Let $_RM\ne 0$ (left $R$-module) such ...
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1
answer
55
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In a unital R-module M with generating set S, there exists a free module with a basis of the same size and an onto homomorphism
Let M be a unital R-module with $S\subseteq M$ a generating set, that is =M, then there exists a free R module F such that with basis B such that:
|S|=|B| and
$\exists \phi \in Hom_R(F,M)$ such that $...
- 153
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vote
1
answer
52
views
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 ...
- 526
1
vote
1
answer
84
views
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 ...
- 33
0
votes
1
answer
67
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The direct sum of modules is generated by their union
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:
$$\bigoplus_{i\in I}A_{i}=\langle \bigcup_{i\in I}A_{...
- 33
1
vote
0
answers
29
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Periodicity of the sequence of ideals generared by the entries of the maps in minimal free resolution of modules over complete intersection ring
For a finitely generated module $M$ over a Noetherian local ring $(R,\mathfrak m)$, let $I_i^R(M)$ denote the ideal generated by the entries in a matrix representation of $\partial_i$, where $(F_i,\...
- 105
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0
answers
40
views
Is it possible to find an algorithm which gives coordinates of the basis elements?
Suppose $R$ to be a ring with unity, and we have a module homomorphism $f: A \to B$, where $A,B$ are $R$-modules.
Suppose the image $f(A)$ is free $R$-module of rank $2$ with $R$-basis $\{u,v\}$. We ...
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1
vote
0
answers
29
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Canonical morphism of inverse limits of inverse systems of modules
Let $A$ be a commutative ring and $I$ an index set. Let $(M_i, f_i)_{I\in I}$ be an inverse system of $A$-modules and let $N$ be an $A$-module. There is a canonical map
$$\phi: (\mathrm{lim}_{\...
- 912
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votes
0
answers
43
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Exercise from Eisenbud — Why is this quotient rank 1?
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 & ...
- 2,292