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

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5
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2answers
167 views

Why is this module reflexive?

Let $A\subset B$ be two integrally closed Noetherian domains with $B$ finitely generated as an $A$-module. Then $B$ is reflexive. Could you explain me why, please? Reflexive means that the ...
6
votes
3answers
218 views

Solutions of homogeneous linear differential equations are a special case of structure theorem for f.g. modules over a PID

In this other post, Qiaochu Yuan comments that the solutions for the homegeneous linear differential equation with constant coefficients are a special case of the structure theorem for finitely ...
8
votes
2answers
864 views

Direct summand of a free module

Let $M$, $L$, $N$ be $A$-modules and $M=N\oplus L$. If $M$ and $N$ are free, is $L$ necessarily free?
6
votes
2answers
321 views

Whence this generalization of linear (in)dependence?

I recently came across a definition of (in)dependence that is supposed to be a generalization of linear (in)dependence among a set of vectors: An element $x$ is dependent on a set of elements ...
14
votes
1answer
324 views

Lemma on infinitely generated projective modules

Is it true that every finitely generated submodule of a non-finitely generated projective over a (not necessarily commutative!) ring is contained in a proper summand? (Ideally there's a standard ...
0
votes
2answers
292 views

A simple module versus a simple ring

In the book I've been reading recently (Algebra by Thomas W. Hungerford) the following definition is given: A left module $A$ over a ring $R$ is simple provided that $RA\neq 0$ and $A$ has no ...
4
votes
1answer
382 views

Calculating Hom(A,B)

I have been studying modules and homological algebra as of late but somehow I have missed how to calculate Hom(A,B) for abelian groups, modules and Hom(A,_)/Hom(_,B) for exact sequences. I have no ...
3
votes
1answer
312 views

Flat algebras and tensor product

All rings are commutative. Suppose $B$ is a flat $A$-algebra, and that $M$ and $N$ are flat $B$-modules. Is there a way to compare the two $A$-modules $M \otimes_A N$ and $M \otimes_B N$? Thanks
0
votes
1answer
257 views

Question about pure submodules over a polynomial ring

I have had trouble coming up with ideas to solving the following problem on pure modules. We have worked a couple of exercises from Jacobson and Hoffman and Kunze on pure modules but this is a sample ...
0
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1answer
410 views

Necessary and sufficient conditions for a module to have finitely many invariant subspaces

I have had trouble answering the following question which is from a study guide to a qualifying exam I will be taking later this summer. I am thinking this question has something to do with cyclic ...
3
votes
1answer
120 views

Counting the number of $F[x]$-submodules

I am trying to answer a couple questions about counting $F[x]$-submodules and I think I am having trouble understanding definitions. I will try to explain how I have been approaching the problem and ...
5
votes
2answers
141 views

Cancellation law in finitely generated modules

Let $R$ be a commutative ring with $1$ and let $S,T$ be $R$-modules such that $S$ is finitely generated and such that $S \cong S \oplus T$. Must $T=0$? This is certainly true if $R$ is a PID, but ...
0
votes
1answer
67 views

Dimensions of modules of the maximal compact subrings of locally compact fields

I have checked the list of similar titles, proposed by the site. I hope this is not a repetition. This question arises from a proof of a proposition in the book Basic Number Theory, as follows. ...
4
votes
1answer
436 views

What are the relations between the Koszul complex and the minimal free resolution?

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and $F.$ the Koszul complex of a minimal system of generators of $\mathfrak{m}$. Let $G.$ be the minimal free resolution of $k$. In which cases they ...
2
votes
1answer
97 views

Isomorphisms of I-adic completions

Background: There is a well-known theorem that if $R$ is Noetherian, $I$ is an ideal in $R$, $0\rightarrow M_1\rightarrow M_2\rightarrow M_3\rightarrow 0$ an exact sequence of finitely generated ...
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vote
2answers
158 views

On the height of an ideal

Which of the following inequalities hold for a ring $R$ and an ideal of its $I$? height $I\leq\mathrm{dim}\;R-\mathrm{dim}\;R/I$ height $I\geq\mathrm{dim}\;R-\mathrm{dim}\;R/I$
1
vote
1answer
309 views

Isomorphism between set of all R-modules homomorphisms

Let $S$ be an $R$-module where $R$ is an integral domain and let $P = \langle p \rangle $ be a prime ideal. Define: $S_{P}=\{s \in S: p^{n} s=0 \ \textrm{for some natural }n\}$. As usual, denote ...
2
votes
1answer
451 views

Finding a basis for a submodule

Let $F$ be the $\mathbb{Z}$ -module $\mathbb{Z}^{3}$ and let $N$ be the submodule generated by {$(4,-4,4),(-4,4,8),(16,20,40)$}, Find a basis $\left\{ f_{1},f_{2},f_{3}\right\}$ for $F\textrm{ and ...
3
votes
1answer
143 views

A simple projective test lemma

Suppose $P$ is a left $M$-module, and suppose that for every injective module $E$, there is a $g:P \to E$ making the diagram commute: $$ ...
4
votes
1answer
210 views

Koszul algebra of a ring

I'm studying on Cohen-Macaulay Rings of Bruns-Herzog. Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and $H_{\bullet}(R)$ its Koszul algebra. I found on the book (page 75) that "since ...
3
votes
2answers
218 views

Auslander-Buchsbaum and Ferrand-Vasconcelos

I'm studying on "Cohen-Macaulay rings" of Bruns-Herzog, here a link: http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false At page 65 there is ...
6
votes
2answers
2k views

Minimal free resolution

I'm studying on the book "Cohen-Macaulay rings" of Bruns-Herzog (Here's a link and an image of the page in question for those unable to use Google Books.) At page 17 it talks about minimal free ...
2
votes
1answer
219 views

Help on a proof of a Theorem of Rees

I'm studying on this book http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false on page 10 there is a Rees Theorem. I'd like to know why the ...
6
votes
1answer
81 views

Why does the ring $A$ need to be commutative for $Hom_A(X,X')$ to be an $A$-module?

Refer to Lang's "Algebra" second paragraph from top, p. 122. Let $A$ be a ring and $X$, $X^'$ be $A$-modules. Let $Hom_A(X,X')$ be the set of $A$-homomorphisms from $X$ to $X'$. It is ...
4
votes
1answer
187 views

Can I assume $R$ is a local ring?

I should prove this statement: Let $R$ be a ring, $M$ be a $R$-module and $P$ a projective $R$-module of finite type. If $x=\sum_i m_i\otimes p_i$ is an element in $M\otimes_R P$ such that $\sum_i ...
2
votes
1answer
196 views

application of the five lemma

suppose we are given a short exact sequence of $\mathbb{Z}G$-modules $$0\to K\to F\to A\to 0$$ where $F$ is free. and we form a diagram with that first row and with a second row $0\to L\to M\to N\to ...
3
votes
1answer
286 views

About presentation of module

Let $R$ be a ring and $R[\mathbb{Z}]$ be the group ring obtained from ring $R$ and group $\mathbb{Z}=<s>$. Suppose that $M$ be a $R[\mathbb{Z}]$-module and it is isomorphic to $R^n$ as ...
2
votes
0answers
139 views

Twisted forms of a free module

Let $R$ be a ring an let $A$ and $B$ be two $R$-algebras. If $S$ is faithfully flat over $R$ then we say that $A$ is a $S$-twisted form of $B$ if $A\otimes_R S$ and $B\otimes_R S$ are isomorphic as ...
2
votes
1answer
135 views

Free presentations of $\mathbb{Z}G$-modules

Dear All, I have a doubt about a specific definition, but I cannot find any help on the web or on the books that I have. Talking about $\mathbb{Z}G$-modules, what does one intend saying "take a free ...
4
votes
3answers
1k views

examples of faithfully flat modules

I'm studying some results about flatness and faithful flatness and I'd like to keep in my mind some examples about faithfully flat modules. In general, free modules are the typical example. Another ...
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vote
3answers
153 views

Are local Artinian algebras injective?

Suppose $R$ is a local Artinian algebra. Question: Is $R$ an injective $R$-module?
0
votes
1answer
61 views

Prove a property of when simple modules are isomorphic

Let $\mathfrak{m}_1$ and $\mathfrak{m}_2$ be left maximal ideals of a unital ring $A$. Show that the simple modules $A/\mathfrak{m}_1$ and$A/\mathfrak{m}_2$ are isomorphic if and only if there exist ...
0
votes
1answer
93 views

Composition factors and annihilation

Let $A$ be an associative algebra over a field $k$ and let $M$ be some $A$-module. If $\mathfrak{a}$ is an ideal of $A$ which annihilates $M$, i.e. if $\mathfrak{a}M=0$, then why must every ...
3
votes
0answers
206 views

group cohomology with coefficient in an induced module

We say that a $G$-module $I$ is induced if $$I\cong L\otimes\mathbb{Z}G$$ where $L$ is an abelian group and the action on $L\otimes\mathbb{Z}G$ is given by the action of $G$ only on the second ...
4
votes
1answer
251 views

Lifting maps of quotient modules

Today I tried to check this, but couldn't see how to do it. I think it is probably a standard result, but a brief check of Atiyah-Macdonald didn't yield anything, and I don't know what to google for. ...
1
vote
1answer
340 views

What is a typical example of the tensor product of modules failing to be left exact?

I am looking for an example of an exact sequence of $R$-modules $$ 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0 $$ and a $R$-module $N$, such that $$ 0 \rightarrow M' ...
2
votes
1answer
483 views

Minimal generating sets of free modules, and endomorphisms of free modules

I know that it seems very loose as a title but I hope this post will be beneficial to all the forum members. One thing I like about free modules is that they help one define maps directly as we do in ...
6
votes
1answer
313 views

Find a f.g. $F[x]$-module which is not projective

I was working through a problem that was showing that for a field $F$, $F[x]$-modules correspond to pairs $(V,T)$ of vector spaces over $F$ and linear transformations on that vector space. The last ...
0
votes
1answer
61 views

$\mathbb{C}$-vector space structure inside nontrivial f.g. module over $M_2 (\mathbb{C})$ has dim $\geq 2$

I'm trying to prove: If $M$ is a nontrivial finitely generated left module over $M_2 (\mathbb{C})$, then the accompanying $\mathbb{C}$-vector space structure (just restrict the action to the scaling ...
3
votes
2answers
100 views

equality of modules

I'm reading a proof of Nakayama's theorem; it says at a certain step that: For $M$, a finitely generated module on a ring $R, N$ a submodule, and $I$ an ideal of the ring $R$: If $M = N + IM$, then ...
1
vote
2answers
122 views

Prove that the group ring $\mathbb{Z}_2[G]$ is not semisimple where $G$ is the cyclic group of order 2

I want to prove that the group ring $\mathbb{Z}_2[G]$ is not semisimple when $G$ is the cyclic group of order two. I guess the simplest way would be to show that there exists a submodule that is not a ...
3
votes
2answers
494 views

elementary question about tensor product of modules

I'm a bit embarrassed to ask this, but I've gotten myself confused over what I think is a simple issue. Let $A$ be a local ring, $k$ its residue field, and $M,N$ finitely generated $A$-modules. An ...
2
votes
1answer
282 views

Name of a book with the following contents?

Some time ago, I received Algebra notes from my advisor who is advising me on a project. I learned very much from these notes and wondered what the name of the book from which these notes were ...
2
votes
1answer
267 views

construction of the tensor product, generating submodule which is factored out

The tensor product of modules $M_0, M_1$ is a quotient of a free module $F$, …, by a submodule $F'$. I found 2 definitions of this $F'$, and the difference is in this generating rules: ...
5
votes
2answers
245 views

For which prime $p$ are the additive groups $\mathbb{F}_p$ and $\mathbb{F}_p^2$ $\mathbb{Z}[i]$-modules?

Homework for my algebra class. Chapter 14, Exercise 7.8 in Artin's Algebra, Second Edition: Let $F = \mathbb{F}_p$. For which prime integers $p$ does the additive group $F^1$ have a structure ...
5
votes
0answers
172 views

Dual modules and first cohomology

Let $G$ be a finite group, $K$ a characteristic-$p$ algebraically closed field (say $p$ divides $|G|$), and let $M$ be a finite-dimensional $KG$-module. What hypotheses are needed on $G$, $M$ to ...
3
votes
1answer
164 views

Endomorphisms of a representation

Let $G$ be a group acting continuously on a free $\mathbb{Z}_p$-module of finite rank. Assume that $End_{G}(T)$ and $End_G(T/p)$ are the homotheties. Is it possible that $End_{G}(T/p^n)$ contains ...
0
votes
1answer
89 views

Superscript wedge on an $R$-module

I'm reading these lecture notes here about the tensor product and the author uses $M^\vee$ where $M$ is an $R$-module. Look at page 2 for an example (example 2.1). What does it mean? I looked through ...
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vote
2answers
68 views

Condition on product implies that is it the trivial module?

Let $M$ be an $A$-module and let $\mathfrak{a}$ be an ideal of $A$. Suppose that $\mathfrak{m} \cdot M =0$ for every maximal ideal $\mathfrak{m}$ of $A$ such that $\mathfrak{a} \subseteq ...
4
votes
1answer
193 views

Elementary question about localization of ideals

I'm trying to show the following: Let $I,J$ be ideals of a commutative ring (with 1) $A$ such that $I_{P}=J_{P}$ for every prime ideal P of R. Here $I_{P}$ means the localization of $I$ in $P$. Then ...