For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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9
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0answers
42 views

Why are noetherian and artinian modules important?

As a TA I was recently asked to give the students an introduction to two (quite related) concepts that are new to me, noetherian and artinian modules. I intend to prove the characterisation theorem ...
-1
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1answer
46 views

Direct sum over ring [closed]

Let $R$ be a PID and $F$ be a free module with basis $\{e_1, e_2\}$. Let $u = me_1 + ne_2$ for $m,n \in R$. Prove that $Ru$ is a direct summand of $F$ if and only if the ideal $(m,n)=R$.
1
vote
2answers
83 views

An abelian group A is torsion iff A ⊗ Q = 0

At the moment I know the fact for finitely generated abelian groups and this: If $x \otimes y = 0 \in M \otimes N$, then there is a finitely generated submodule $L \subset M$ containing $x$ such that ...
0
votes
0answers
19 views

Any cyclic module can be given an algebra structure?

Given a cyclic module $M$ over a commutative ring $R$, we know that $M$ is isomorphic (not canonically) as an $R$ module to $R/A$, where $A$ is the annihilator of $M$ in $R$. But $R/A$ is an $R$ ...
1
vote
1answer
24 views

Direct sum of free module?

Let $R$ be a PID, and let $F$ be a free module with basis $e_1, e_2$. Let $u\in F$ such that $u = me_1 + me_2$ where $m \in R$. How can I show that $Ru$ is a direct summand of $F$ if and only if $m$ ...
2
votes
1answer
50 views

Extending scalars from $k$ to $K$: how find $K$ linear maps?

If I extend scalars from a ring $R$ to a ring $S$ by a homomorphism $f:R \to S$, then starting with an $R$ module $M$, I get an $S$ module $S \otimes_R M$. Given $\sigma \in \text{End}_R(M)$, I know ...
5
votes
2answers
99 views

Proof of the properties of tensor product

On page 25 of Atiyah-Macdonald "Introduction to commutative algebra", the author says that "We shall never again need to use the construction of the tensor product given above and the reader may ...
0
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0answers
25 views

How do I generalize this theorem?

Dummit&Foote p.362 Let $R$ be a subring of $S$ and let $N$ be a left $R$-module and let $\iota:N\rightarrow S\otimes_R N:n\mapsto 1\otimes n$ be an $R$-module homomorphism. Let $L$ be ...
0
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0answers
18 views

What's the point of the extension of scalars?

Let $R,S$ be rings and $f:R\rightarrow S$ be a ring homomorphism. Then we give $S$ a right $R$-module structure as $s•r =sf(r)$. It seems natural to give an $R$-module structure in this way, but I ...
0
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2answers
32 views

Example to show that multiplication by ideals and intersection of submodules do not commute

The key point of question about typical proof of Krull Intersection Theorem is that multiplication by ideals and intersection of submodules do not commute. Can anyone give me an example of this? ...
0
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0answers
30 views

Does the abelian group in the definition of a left R-module need to be additive?

Let $R$ be an arbitrary ring and $M$ an abelian group, a left $R$ module consists of the aforementioned abelian group $M$ and an operation $f:R \times M\rightarrow{M}$ satisfying for $r,s\epsilon{R}$ ...
0
votes
1answer
22 views

Example of module homomorphism?

I would like to come up with an example for a module homomorphism between two finitely generated, not free modules over $\mathbb{Z}[i]$. Also, what would be the kernel and image in this example?
1
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1answer
57 views

Intersection of submodules

I have question regarding intersection of submodules. Could anyone give example of a commutative ring $R$ with identity and an $R$-module $M$ such that $$IM\cap JM\nsubseteq (I\cap J)M$$ for some ...
3
votes
1answer
62 views

Show an ideal is a finitely generated projective module via a split exact sequence

Let $I$ be an ideal of $R$ such that the mapping $f:I\otimes_R\operatorname{Hom}_R (I,R)→R$ defined (on the generators) by $f(i\otimes α)=α(i)$ for all $i∈I$ and $α∈\operatorname{Hom}_R (I,R)$ is ...
2
votes
0answers
43 views

$\mathbb{Z}_p$ as a module over $\mathbb{Z}_{(p)}$

I denote by $\mathbb{Z}_{(p)}$ the localization at a prime p, and by $\mathbb{Z}_p$ the p-adic integers. Question: what is the structure of $\mathbb{Z}_p$ as $\mathbb{Z}_{(p)}$- module? For example ...
2
votes
3answers
94 views

Does a long exact sequence of flat modules remain exact after tensoring with an arbitrary module?

In Liu's Algebraic Geometry and Arithmetic Curves, Proposition 1.2.6 states that given any short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ with $M''$ flat, taking ...
1
vote
1answer
28 views

Direct sum is isomorphic?

Given R, a PID, and two finitely generated R-modules A and B, suppose $\varphi: A \rightarrow B$ is a homomorphism. If B is a free module, show that $A \cong \ker(\varphi) \oplus ...
0
votes
0answers
48 views

A property for finitely generated modules

Consider the folowing property for an $R$-module $M$ (where $R$ is an associative ring with identity): For each family $\{M_{\alpha}\}_{\alpha\in I}$ of $R$-modules and each homomorphism $f:M\to ...
0
votes
0answers
18 views

Closure properties for classes of modules that form a cotorsion pair

A torsion theory is a pair of classes of $R$-modules (where $R$ is an associative ring with identity) $({\mathbb T},{\mathbb F})$, such that $r({\mathbb T})={\mathbb F}$ and $l({\mathbb F})={\mathbb ...
2
votes
1answer
41 views

divisible modules over Dedekind Domains

L. Fuchs in one of his articles says that: "divisible modules over Dedekind Domains can be completely characterized by numerical invariants". Please introduce me to a source in this respect. I so ...
1
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1answer
16 views

Superfluous condition on universal property of tensor product

Let $R$ be a ring and $M$ be a right $R$-module and $N$ be a left $R$-module. Let $\phi:M\times N \rightarrow M\otimes_R N$ be the canonical middle linear map. Here is a theorem in my text under the ...
2
votes
1answer
61 views

Equivalence of categories involving graded modules and sheaves.

Let $S$ be a graded ring with $S_0=A$ a finitely generated $\mathbb{K}$-algebra and $S_1$ a finitely generated $A$-module. Let $M$ be a graded $S$-module and $\tilde{M}$ the corresponding sheaf on ...
0
votes
0answers
11 views

Characterisation of module structure on the tensor product

Let $S,R$ be rings and $M$ be an $(S,R)$-bimodule and $N$ be a left $R$-module. Then, there exists an openration $•:S\times (M\otimes_R N)\rightarrow M\otimes_R N$ such that $s•(m\otimes n)=sm\otimes ...
1
vote
1answer
38 views

An inverse limit exact sequence for complete modules

Let $A$ be a commutative complete ring with unit for the $I$-adic topology, where $I$ is the ideal of $A$. Let $(M_n)_{n\geq 0}$ be $A$-modules such that $I^{n+1}M_n=0$ and that there exist a ...
0
votes
2answers
54 views

Quotient Module is Not Free

Let R be a commutative ring with a multiplicative identity. $a\in R, a \not = 0$, prove that $R/(a)$ is not a free R-module. My initial attempt was to say that R/(a) is not torsion free, thus it ...
3
votes
1answer
52 views

Quotient of modules

I' m having trouble with the following exercise: Let $Y$ be a submodule of $X$ and $X$ a submodule of $Z$ (modules over a ring $R$). Suppose that $X/Y$ is isomorphic to $Z$. I need to conclude ...
7
votes
1answer
122 views

How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace. I ...
1
vote
1answer
46 views

If a module homomorpic image is free, then does a module in domain free?

Let's say we have $f:A \to B$ where $A$ and $B$ are R-modules and $f$ is a R-module homomorphism. If A has a torsion, then $\{a \in A \mid ar=0 \text{ for some nonzero } r \in R\}\neq \varnothing ...
0
votes
2answers
36 views

finite dimensional $K$-vector space $V$ with linear endomorphism $T$ is a cyclic module over $K[x]$

Problem: Suppose that $V$ is a finite dimensinal $K$-vector space with a linear endomorphism $T$. Then show that i) Associated $K[x]$-module $V$ is such that $V\cong K[x]/(g)$ for some monic ...
1
vote
2answers
42 views

Does tensoring the integers $\Bbb Z$ with any field $F$ give that field itself? How to prove that?

Does tensoring the integers $\Bbb Z$ with any field $F$ give that field itself? How to prove that? I know for $\Bbb Q$ it is true.
1
vote
1answer
12 views

Definition of the tensor product of finite sequence of modules

I have posted several questions about the tensor product of modules before and this post would be the final one. I have read wikipedia,mathSE,Dummit&Foote and Bourbaki for the definition of the ...
0
votes
1answer
61 views

If $V$ is a vector space to itself, is the idempotent linear transformation, prove there is a basis such that is a diagonal matrix with all $0$ or $1$

Im trying to solve an algebra homework problem, basically i have: $$\phi:V \longrightarrow V$$ such that $\phi = \phi^2$, where $V$ is finite dimensional. I already proved that $V = \phi(V) \oplus ...
0
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0answers
16 views

Definition of multimodule

Reference : Bourbaki - Algebra I p.224 Let $\{A_i\}_{i\in I}$ and $\{B_j\}_{j\in J}$ be families of rings. Let $M$ be a set such that for each $i\in I, M$ is a left $A_i$-module and for each ...
0
votes
1answer
42 views

Faithfully Flat Abelian Groups

I need some help to find faithfully flat abelian groups. Flat abelian groups are torsion free $\mathbb{Z}$-modules. But what about faithfully flat abelian groups. $\mathbb{Q}$ is an example that is ...
1
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0answers
28 views

Simplified Exponent in $\bmod$ equation

I am trying to simplified the following expression: $$\left(y^m\right)^{x} \bmod p$$ In my case, I can only solve $(y^x) \bmod p$ first without prior knowledge of $m$. Eventually, my answer should ...
1
vote
1answer
30 views

Definition of tensor product

Here is the standard dedonition of the tensor product of two modules: Definition for tensor products of two modules: Let $R$ be a ring and $M$ be a right module and $N$ be a left module. ...
1
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0answers
26 views

Is there less use of infinite rank modules than finite rank modules?

Let $R$ be a nonzero ring and $M$ be a free module on $R$ and here are theorems I know. If $M$ has a infinite basis, then rank of $M$ is well-defined If $R$ has IBN property, then rank of $M$ ...
0
votes
0answers
36 views

What is $\sum dim(H_i(X,F))$?

Let $X$ be a finite CW complex and $F$ be any field. Then what is dimension of $\sum_iH_i(X,F)$? This question has three sub parts as far what I think i) if $F$ is of char $0$ then $\Bbb Q \subseteq ...
2
votes
0answers
38 views

indecomposable summand

Let $M$ be an $R$ module. Is this true for start a proof that we say "Let $S$ denote the indecomposable summands of $M$"? In fact, I want to know whether any module over a Dedekind domain (or a ...
0
votes
1answer
31 views

What are the generalized statements of given statements?

Let $R$ be a ring and $M$ be an $R$-module. Thm1. If $R$ is a division ring, then $M$ is free Thm2. If $R$ is commutative, the rank of $M$ is unique. First of all, are these statements ...
0
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0answers
26 views

Embedding a ring in a direct product

If an $R$-module $C$ is a homomorphic image of a direct sum $⊕M$, where $M$ is an $R$-module, and $R$ could be embedded in a direct product $ΠC$, could $R$ be embedded in a direct product $ΠM$?
0
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0answers
27 views

How to compute cyclic/dihedral powers of modules?

The $n$-th cyclic power of an $R$-module $M$ is $$T^n_C(M)=M^{\otimes n}/\langle m_1\!\otimes\!m_2\!\otimes\!\ldots\!\otimes\!m_n-m_2\!\otimes\!\ldots\!\otimes\!m_n\!\otimes\!m_1;\, ...
5
votes
2answers
341 views

Is a finitely generated projective module a direct summand of a *finitely generated* free module?

Let $R$ be a (not necessarily commutative) ring and $P$ a finitely generated projective $R$-module. Then there is an $R$-module $N$ such that $P \oplus N$ is free. Can $N$ always be chosen such that ...
1
vote
0answers
18 views

Does faithfully flat descent work using restriction of scalars rather than extension?

Vistoli's notes on fibred categories and descent - http://homepage.sns.it/vistoli/descent.pdf - introduce (section 4.2.1) descent on modules over a commutative ring. The idea is as follows: ...
3
votes
1answer
60 views

Coker of powers of an F in End(M)

If Coker of an $F$ in End$(M)$ is of finite length, where $M$ is Noetherian, is Coker and Ker of all powers of $F$ of finite length? Is the condition of being noetherian necessary?
0
votes
0answers
32 views

Projectivity of a certain module

Let $M$ be a generator for the category of left $R$-modules, and let we have an $R$-epimorphism $h$ from $R^{(X)}$ to an $R$-module $P$ which is projective relative to $M^{(X)}$ ($X$ is a set). I want ...
0
votes
1answer
20 views

Cauchy sequence in a valuation ring

From Janusz's book algebraic number fields, chapter 2. Let $K$ be a complete field with respect to a non-Archimedean absolute value $|\cdot|$. Let $R$ be its valuation ring with maximal ideal ...
2
votes
1answer
24 views

What is the standard notation for the submodule generated by a set?

In group theory, the standard notation for the subgroup generated by a set is $<A>$. In ring theory, the standard notation for the ideal generated by a set is $<A>$ while there is no ...
1
vote
1answer
32 views

Torsion module decomposition into cyclic factors.

I am trying to resolve some confusion about the decomposition of a torsion module into cyclic factors. Let $R$ be a PID. Let $M$ be a finitely generated torsion $R$-module. That is, $M=T(M)=\{m \in ...
1
vote
1answer
35 views

When is a pure tensor equal to $0$?

Let $R$ be a commutative unital ring and let $A$ and $B$ be $R$-modules. For any $b\!\in\!B$ the set $$\{a\!\in\!A;\, a\!\otimes\!b\!=\!0\text{ in }A\!\otimes_R\!B\}$$ is a submodule of $A$ that ...