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

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7 views

Tensor product of infinite-dimensional Hilbert spaces and tensor product of $\mathbb{C}$-modules.

In here I found the following construction. Let $R$ be a commutative ring, and $M,N$ be $R$-modules. The set $M\times N$ is well defined, and it is the starting point of the definition of the tensor ...
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1answer
20 views

Flatness via factoring homomorphisms

Theorem (4.32) of "Lectures on modules and rings" by T.Y. Lam says that a module $P_R$ is flat iff any $R$-homomorphism $λ:M→P$ where $M$ is any finitely presented $R$-module can be factord ...
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16 views

Is any Direct Summand of a Free Module over a PID Also Free?

Let $R$ be a PID and $F$ be a free module over $R$. Suppose we have $F=A\oplus B$ for some $R$-modules $A$ and $B$. Then are $A$ and $B$ necessarily free? If $F$ is finitely generated, then I ...
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1answer
11 views

Definition of linear independence in R-module

I am revising module theory over commutative rings with 1 and I have a "soft" question regarding to why don't we define linear independence as follows: "$v_1,...,v_n$ are linearly independent if ...
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0answers
47 views

What do we call the ring homomorphism $R \rightarrow \mathrm{End}_{\mathbf{Ab}}(X)$ associated with an $R$-module $X$?

First, a convention: given an abelian group $X$, write $\mathrm{End}_{\mathbf{Ab}}(X)$ for the set of all group homomorphisms $$X \rightarrow X.$$ Now let $R$ denote a ring. Question. Given an ...
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1answer
54 views

Need a counterexample: If the induced homomorphism $M/\mathfrak aM \to N/\mathfrak aN$ is surjective, then $f$ it's surjective.

This problem is from Atiyah and Macdonald, Introduction to Commutative Algebra, Exercise 10, Chapter 2. Let $A$ be a commutative ring with $1 \ne 0$ and let $\mathfrak a$ be an ideal of $A$ ...
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47 views

Find a polynomial equation satisfied by $\phi$

I'm solving the following exercise from my class notes: Let $A=k[x^2,y^2]$ and let $M=k[x^2,xy,y^2]$ ($k$ a field). Show that $M$ is an $A$-module. Define $ \phi :M \to M$ by $\phi(m)=xym$. Find ...
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2answers
39 views

Is $V$ a simple $\text{End}_kV$-module?

Let $V$ be a finite-dimensional vector space over $k$ and $A = \text{End}_k V$. Is $V$ a simple $A$-module?
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1answer
40 views

Injective hull of a simple module

Let $M$ be an indecomposable injective right module over a right Artinian ring $R$, so $M$ has exactly one associated prime ideal $P$ (Lectures on Modules and Rings, T.Y. Lam). Now, $R/P$ is a simple ...
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33 views

Projective dimension over a factor ring

$\newcommand{\pdim}{\operatorname{pdim}}$If $\pdim_A M$ is the projective dimension of $M$ as an $A$-module how can i prove that if $A/I=A'$ then $$\pdim_A M\leq \pdim_A A' + \pdim_{A'} M$$ If the ...
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1answer
30 views

Security of such cryptosystem design?

Is one able to reveal $m$ when $$С = (m + r)^e \bmod N$$ $C$ is known $r$ is known $e$ is known $N$ is known and not prime
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1answer
55 views

Finite Modules Isomorphism

Two vector spaces are isomorphic if and only if they have the same dimension. In particular, two vector spaces over a finite field are isomorphic if and only if they have the same cardinality. For a ...
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1answer
28 views

Is the inclusion map always a module homomorphism?

Suppose $R$ is some commutative ring and $G$ a finite group so that $R[G]$ is the usual group ring. If $M$ is some $R[G]$-module, then we can inject $M\hookrightarrow M\oplus R[G]^n$ for some ...
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1answer
34 views

Does there exist nonzero eigenvector such that $E(v) = 0$?

This is a followup to my previous question here. Let $\text{U}_+$ be an associative $\mathbb{C}$-algebra with two generators $E$, $H$, and one defining relation $HE - EH = 2E$. Let $M$ be a ...
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0answers
44 views

If $M$and $N$ are R- modules, then under what conditions $\operatorname{Hom}(M,N)$ the space of R-module morphisms from M to N, is projective?

If M and N are R- modules, then under what conditions $\operatorname{Hom}(M,N)$ is projective? I was trying to show that Hom(M,N) might be written as tensor product of two modules, i.e. Of dual of M ...
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2answers
31 views

$M \neq 0$ but $M^* = 0$.

Let $A$ be a ring. For any left, resp. right $A$-module $M$ give the abelian group $\text{Hom}_A(M, A)$ the structure of a ring $A$-module (to be denoted $M^*$), resp. left $A$-module (to be denoted ...
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1answer
47 views

Are all rings $\mathbb{Z}$-modules?

In my course of associative algebra we covered modules and an excercise involved showing that every ring $R$ can also be viewed as an $R$-module. This was straightforward enough. Is it also true that ...
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0answers
30 views

$M \otimes_\mathbb{C} N$ is simple.

Let $A$ and $B$ be finitely generated $\mathbb{C}$-algebras. For any simple modules $M$ and $N$, over $A$ and $B$ respectively, how do I see that the $A \otimes_\mathbb{C}B$-module $M ...
3
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1answer
38 views

Any two $A$-modules of the same dimension over $k$ isomorphic as $A$-modules?

Let $A$ be a central simple $k$-algebra. Are any two $A$-modules of the same dimension over $k$ isomorphic as $A$-modules?
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1answer
17 views

Show that a one dimensional $\mathfrak {g}\!-\!\operatorname{module}$ is irreducible

I want to show that a one dimensional $\mathfrak {g}\!-\!\operatorname{module}$ is irreducible. My problem comes down to undestanding the basis of this object. Now assuming the basis works in the ...
2
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1answer
37 views

What is the injective envelope of a product of abelian groups?

We know that any $R$-module has an injective envelope. Matlis has shown that any injective module over a Noetherian $R$ decomposes as a direct sum of indecomposable injectives, and that these are the ...
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0answers
17 views

Homomorphisms of twisted modules

Let $A$ and $B$ be $R$-bimodules, and $\alpha$ an $R$-automorphism. Write ${}_1A_\alpha$ for the right-twisted $R$-bimodule with action $(r\cdot x\cdot s)\mapsto rx\alpha(s)$. Is it true that a ...
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38 views

a problem in homological algebra

For $C$ is abelian group satisfied $pC=0$ with p is a prime number and $G$ is abelian group. prove that $Ext_{Z}(C,G)\cong Hom(C,G/pG)$ I thought about this in 2 hours but couldn't prove it!
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24 views

Composition of module homomorphisms and their kernels

Let $R$ be a principal ideal ring and let $I$ and $J$ be two ideals of $R$. Suppose $\phi: R \times R \rightarrow R \times R$ and $\psi: R \times R \rightarrow R \times R$ are two $R$-module ...
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1answer
39 views

What is $Ext_{\mathbb Z}^{1}(S^1, \mathbb Z)$?

By wikipedia, suppose $A, B$ are left $R$-modules, one way to calculate $Ext_{R}^{1}(A, B)$ is to regard it as equivalent class of module extension of $A$ by $B$, in the sense that the diagram ...
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0answers
45 views

Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. [closed]

I took this exercise for a long time but I can't prove it. Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. Anyone could help me?
2
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1answer
27 views

Integrality Test: Matrices as Homomorphisms of Modules

I am reading in Neukirch's Algebraic Number Theory book and I find an argument which I stumble upon in several places and which I do not understand. The discussion is fairly at the beginning and the ...
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0answers
37 views

Primitive vectors in $A^n$

Let $A$ be a commutative ring with 1. Let n be a positive integer. I call a vector $(a_1,...,a_n) \in A^n$ primitive, if the ideal generated by $\{a_1,....,a_n\}$ is $A$. Question: Given a primitive ...
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23 views

Module endomorphisms with the same kernel

Let $R$ be a finite commutative principal ideal ring. Let $n$ be a positive integer. For $i=1, \ldots, n-1$ we let \begin{align*} w_i := w_i(x_{i+1}, \ldots, x_n) = \sum_{j=i+1}^n t_{ij}x_j \in ...
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0answers
28 views

Lemma 3.53 Rotman's “An Introduction to Homological Algebra”

In Rotman's book An Introduction to Homological Algebra, in the proof of Lemma 3.53, I think we don't need the fact that $\mathbb Q/\mathbb Z$ is injective, because we don't have $0$'s in the ...
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0answers
15 views

Local Constancy of Rank Function

Recently I asked this question. I believe that I have come up with a solution, but I am unsure, because the proof I have seems too easy to be true, and doesn't make very many assumptions. My ...
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2answers
71 views

If a chain complex is homotopy equivalent to its homology, is it split?

Setup and conventions: Let $C_*$ be a chain complex of $R$-modules over some ring $R$, with boundary map $d$. The chain complex is said to be split if there exist $R$-linear maps $s: C_*\rightarrow ...
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2answers
91 views

Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?

Is easy to show that $(\mathbb{R},\mathbb{Q},+,\cdot)$ is isomorphic to $(\mathbb{R}^n,\mathbb{Q},+,\cdot)$ as vectorial spaces and then $(\mathbb{R},+)$ isomorphic to $(\mathbb{R}^n,+)$. This result ...
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1answer
59 views

Show that $1\otimes (1,1,\ldots)\neq 0$ in $\mathbb{Q} \otimes_{\mathbb{Z}} \prod_{n=2}^{\infty} (\mathbb{Z}/n \mathbb{Z})$.

Show that $1\otimes (1,1,\ldots)\neq 0$ in $\mathbb{Q} \otimes_{\mathbb{Z}} \prod_{n=2}^{\infty} (\mathbb{Z}/n \mathbb{Z})$. Here's what I tried: If $1\otimes (1,1,\ldots)= 0$, then $1\otimes ...
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0answers
10 views

Free modules on two isomorphic sets are also ismorphic

Consider the definition of free module in terms of mapping properties. Consider $X_1$ and $X_2$ are two isomorphic R-modules. If $F_{X_1}$ and $F_{X_2}$ are two free modules on $X_1$ and $X_2$ ...
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0answers
21 views

Relation between simple and indecomposable modules

I know that every simple module is indecomposable but the reverse case isn't true in general. So is there a way or conditions that we can add on the indecomposable modules so that they are a simply ...
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1answer
35 views

Functor $M\otimes_B -$ is left and right adjoint to $\operatorname{Hom}_A(M,-)$ when there is a symmetrizing form for $A$?

Suppose $A$ is an $R$-algebra over a commutative ring $R$, which is finitely generated and projective as an $R$-module. A symmetrizing form is a map $t\in\operatorname{Hom}_R(A,R)$ such that ...
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2answers
62 views

What explicitly is the “adjunction” isomorphism $Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))$?

Suppose $B$ and $C$ are commutative rings, $A$ a $B$-algebra, and $B$ is a $C$-module. What exactly is the "adjunction" isomorphism $$Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))?$$ Given $A\to C$, it needs to ...
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1answer
20 views

Is the homotopy category of modules over a quasi-Frobenius ring (pre)additive?

Let $R$ be a quasi-Frobenius ring and $Mod_R$ the category of $R$-modules. One can prove that it admits a model structure whose weak-equivalences are stable equivalences, whose cofibrations are monos ...
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1answer
53 views

Does $f\otimes_A 1_{A/m}:M\otimes A/m\to N\otimes A/m$ injective for all maximal $m$ imply $f$ is an isomorphism?

Let $A$ be a commutative ring. Suppose $f\colon M\to N$ is a morphism of free $A$-modules of equal, finite rank. If $f\otimes_A 1_{A/m}$ is injective for all maximal ideals of $A$, does this imply ...
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0answers
35 views

Making sense of multiplication in tensor product

Let $\mathbb{Z} / p$ and $\mathbb{Z} / q$ be $\mathbb{Z} /pq$ modules where $p$ and $q$ are distinct primes. Then $\mathbb{Z}/p \times \mathbb{Z}/q$ is a $\mathbb{Z} / pq$ module Now let $$ i : ...
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1answer
24 views

A direct sum of cyclic modules over a PID

Let $R$ be a PID, let $p$ be a prime of $R$, and let $M$ be the $R$-module $R/Rp^{e_1}\oplus \cdots \oplus R/Rp^{e_n}$ where the $e_i$ and $n$ are positive integers. Define $M(p)=\{m: pm=0\}$ and ...
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0answers
19 views

If $M$ is an $A$-module via $\varphi\colon A\to\operatorname{End}(M)$, $\mathrm{coker}(\varphi)\otimes M$ is projective?

I have a brief passage I don't understand. Suppose $R$ is a commutative ring, $A$ is an $R$-algebra, which is projective and finitely generated as an $R$-module. Let $M$ be a progenerator for $A$, so ...
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1answer
24 views

Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$.

Let $R$ be a local commutative ring with the maximal ideal $M$. Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$. I tried to apply ...
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1answer
24 views

Finite ring extension of local rings

Let $R$ and $S$ be local rings with the maximal ideals $M$ and $N$, respectively. Assume that $R\subset S$ and that $S$ is a finitely generated $R$-module. If there exists a proper ideal $I$ of $R$ ...
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0answers
36 views

Module isomorphisms and coordinates modulo $p^n$

Let $p$ be a prime and $n \in \mathbb{N}$ is such that $p^n > 2$. We let $\alpha \in \mathbb{N}$ be such that $0 < \alpha < n$. Let $R := \mathbb{Z}_{p^n}$ denote the ring of integers modulo ...
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2answers
53 views

Can we construct a homomorphism from a projective module into a free module?

In short, I have a projective module and a free module, and want to construct a module homomorphism between the two. Is this always possible, at least in some way? Let me go into more detail. Suppose ...
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0answers
16 views

Regarding Adjugate of a Module Homomorphism

$\newcommand{\adj}{\text{adj}}$ I am trying to understand the concept of adjugate of an endomorphism of free module of finite rank as discussed here in Section 8. Let $M$ be a free module over a ring ...
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1answer
40 views

Let $R$ be a commutative domain with field of fractions $F$. Prove that $F$ is an injective $R$-module.

Let $R$ be a commutative domain with field of fractions $F$. Prove that $F$ is an injective $R$ module. I have tried to apply Baer's criterion: every $R$-module homomorphism from any ideal $I$ of ...
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1answer
30 views

Prove that if $M$ is a simple $R=k[x_1,…,x_m]$ -module, then the dimension of $M$ over $k$ is finite.

Let $k$ be a field and let $R=k[x_1,...,x_m]$ be the polynomial ring in $m$ indeterminates. Prove that if $M$ is a simple $R$-module, then the dimension of $M$ over $k$ is finite. I think since ...