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

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10
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222 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we ...
9
votes
0answers
59 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 ...
9
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248 views

An elegant description for graded-module morphisms with non-zero zero component

In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the ...
7
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57 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
7
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0answers
332 views

Condition for a ring on projective and free modules problem

Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is ...
6
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208 views

Elementary divisors theorem for Dedekind domains (Exercise in Lang's Algebra)

Exercise 13 (b) of Chapter III in Lang's Algebra is as follows. Let $M$ be a finitely generated projective module over the Dedekind ring $\mathfrak{o}$. Then there exists free modules $F$ and ...
6
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79 views

Symmetric Algebra and Extension of Scalars

I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation $$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to ...
6
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268 views

Proof of Smith Normal Form as a Generalization of Rank-Nullity Theorem

For any matrix A with entries in a PID, there exist invertible matrices P and Q such that B = PAQ, where B is in Smith normal form. This theorem is usually proved by using elementary row/column ...
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41 views

Exists an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?

Let $M$ be an $R$-module with some torsion element $m$. Does there exist an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?
5
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165 views

A few questions about a specific ring

My question is kinda long, so please bear with me... And I only need hints to get me started. So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & ...
5
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102 views

Category of Chain Complexes of $R$-modules

So I have a couple of questions: 1- Formally speaking, what is a "quotient of a chain complex" of $R$-modules? 2- I want to show that any chain complexes of $R$-modules $C_\bullet $ is the ...
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98 views

Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
5
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181 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 ...
4
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47 views

Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.

See image below. I just want help proving that all kernels in $R$-Mod are monics. My attempt: Let $f : A \to B$ be a map in $R$-Mod. Suppose $i$ is a kernel of $f$, that is: $fi = 0$ and ...
4
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61 views

If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?

Let $A$ be a local (not necessarily noetherian) ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Let $M$ be a finitely generated $A$-module such that $\mathfrak{m}\otimes_A M\rightarrow ...
4
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54 views

When does the duality functor commute with the wedge power functor?

When working with modules over a fixed commutative ring, I know that $(M \otimes N)^* \cong M^* \otimes N^*$ provided either $M$ or $N$ is finitely generated projective. Does it follow that ...
4
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78 views

Property of free submodules for a module over a PID

It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is not free. We can take $R=M=K[x,y]$ , $L=<x>$ , ...
4
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158 views

The dimension of vector space $F^{X}$.

Here's the problem: Let $F$ be field, $X$ an infinite set and $F^{X}$ be the set of all functions $f:X\rightarrow F$. Then $F^{X}$ is a vector space over $F$ (with $(f+g)(x)=f(x)+g(x)$ and ...
4
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0answers
52 views

On semisimple rings

Let $R$ denote a ring with unity. I know that, if $R$ is semisimple, then every $R$-module is semisimple. In particular the class of indecomposable $R$-modules coincides with the class of simple ...
4
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49 views

Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
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66 views

Commutativity of direct and inverse limits

In exercise 5.34(iv) of Homological Algebra book by Rotman one is asked to prove that direct limits and inverse limits do not necessarily commute. I have two questions : 1.) Is it true that ...
4
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68 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
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133 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
4
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68 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
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112 views

The importance of being Cohen-Macaulay

I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book. In that book there are many examples and sentences of the type "If something satisfies this properties, then it is ...
4
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69 views

Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
4
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0answers
120 views

Finitely presented modules and torsion

Suppose $R$ is a Dedekind domain and $A$ a $(m \times n)$-matrix of rank $r$ over $R$. $A$ induces an $R$-module homomorphism $\varphi:R^m \to R^n$ via $x \mapsto xA$ giving rise to an exact sequence ...
4
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0answers
183 views

When is the canonical extension of scalars map $M\to S\otimes_RM$ injective?

Let $\alpha:R\to S$ be a map of unital rings, and let $M$ be an $R$-module. We have a canonical map of $R$-modules: ...
4
votes
0answers
69 views

Decomposing the group algebra $\mathbb{R}[Q]$ as a product of matrix rings

I am trying to decompose $\mathbb{R}[Q]$ as a product of matrix rings ($Q$ is the group of quaternions) By Maschke's theorem, $\mathbb{R}[Q]$ is semisimple. I will begin by decomposing it as a left ...
4
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161 views

Comparison of positive elements and Hilbert C*-modules

I can't find a proof of facts like the following, which apparently are quite standard in the theory of C*-algebras. Let $\mathfrak A$ be any C*-algebra, and $a,b$ two positive elements in $\mathfrak ...
4
votes
0answers
28 views

An explicit $\Lambda_R^\ell(M)$ when $M$ is not free

Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated. When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation} \Lambda_R^n(M)\cong ...
4
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84 views

Separability of finitely generated projectives over commutative ring

A class $\mathcal{C}$ of $R$-modules is called -separative if $A \oplus A \simeq A \oplus B \simeq B \oplus B$ implies $A \simeq B$ for each $A,B \in \mathcal{C}$ -cancelative if $A \oplus C \simeq ...
4
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41 views

Normal Form problem (in a module over a PID)

Let $A,B$ be $n\times n$ matrices with entires in a PID $D$ and $\det AB\neq 0$. Suppose diag$\{a_i\}$, diag$\{b_i\}$, and diag$\{c_i\}$ are normal froms for $A$, $B$, and $AB$. In particular, ...
4
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0answers
359 views

On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
4
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121 views

Ext in Dedekind domains

I know and can prove that $\operatorname{Ext}_Z^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/pA$. Does similar formula work for more general rings, such as Dedekind domains and their ideals, i.e. ...
4
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251 views

Zero divisors in modules?

Let $R$ be a ring. I find myself considering $M=R^n$, as an $R$-module. If $a$ is not a zero divisor in $R$, it holds that $\forall x \in M: ax=0 \Rightarrow a=0 \vee x=0$ . For what kinds of ...
4
votes
0answers
86 views

Decompose a module $M$ of the form $N \times N$, where $N$ is simple

Let $M$ be a $\mathbb{C}[G]$-module of the form $M=N\times N$, where $N$ is simple. How to conclude that $M$ has infinitly many direct sum decompositions into two copies of $N$ ? This is what I have ...
4
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0answers
129 views

Why has the category of all discrete $G$-modules not enough projectives when $G$ is profinite?

I found this article about Galois Cohomology. In it, it says that, when $G$ is a profinite group and $\mathbf{C}_G$ is the category of all discrete $G$-modules, $\mathbf{C}_G$ doesn't have enough ...
3
votes
0answers
12 views

Computing simplicial homology via Smith Normal Form over Rings

I am not sure whether this is the right forum to ask such a question, if not please let me know. In the context of my masters thesis, I am working on writing a program to compute simplicial homology ...
3
votes
0answers
21 views

Right projective but not bimodule projective

Let $R$ be a ring with $1$. What is an example of a right $R$-module $M$ (and a ring $R$) such that $M$ is projective as a right $R$-module but not projective as an $R$-bimodule (assuming $M$ has ...
3
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0answers
44 views

How much information about $R-\mathrm{Mod}$ can be extracted from $\underline{R-\mathrm{Mod}}$ and $K_0(R)$?

The question is in the title, so let me just repeat it: How much information about $R-\mathrm{mod}$ can be extracted from $\underline{R-\mathrm{mod}}$ and $K_0(R)$? Here ...
3
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0answers
57 views

Every module over a division ring is free?

I am currently trying to answer the following true/false question: True or False: Every module over a division ring $R$ is free. I know the result would be true if $R$ is a field (ie a ...
3
votes
0answers
57 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions ...
3
votes
0answers
71 views

finite field extensions: how to compute norm and trace

I'm studying abstract algebra and I'm stuck in the topic of fields. I don't understand what the following definition Let $R$ be a commutative ring and let $S$ be a commutative $R$-algebra, which is ...
3
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0answers
43 views

Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
3
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84 views

If $R$ is a domain and $M$ a finitely generated $R$-module, is it true that $\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M$?

Let $R$ be a domain and $M$ a finitely generated $R$-module. Let $M^{*}=\hom_{R}(M,R)$. Let Tor$M$ be the torsion submodule of $M$. It it true that $$\displaystyle\bigcap_{f\in ...
3
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88 views

$\operatorname{supp}(M) \subseteq \operatorname{supp}(N) \iff f_I(M)\subseteq f_I(N) $?

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. It has proven (here) that if $\operatorname{supp}(M) \subseteq \operatorname{supp}(N)$ then ...
3
votes
0answers
101 views

Length of a composition series of a module

If $A=\mathbb{C}[x,y]_{(x,y)}$, then what is the length of $A$-module $$A/(x^3-x^2y^2+y^{100},x^3-y^{999})\ ?$$ Any suggestion ?
3
votes
0answers
69 views

Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
3
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0answers
88 views

When is $N\otimes_A B \to N$ an isomorphism?

Let $A, B$ be commutative (unital) rings and $f\colon A \to B$ an $A$-algebra. There then exists a canonical functor $f_*\colon \mathbf{Mod}_B \to \mathbf{Mod}_A$ such that, for every morphism of ...