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

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42 views

What do we call the function $R \rightarrow \mathrm{End}_{\mathbf{Set}}(X)$ 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 ...
2
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0answers
30 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 ...
2
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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
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
39 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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1answer
52 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 ...
2
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1answer
50 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 ...
2
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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?
3
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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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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
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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39 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
442 views

Motivation behind the definition of localization

What is the motivation behind definition of localization of rings? From where does the term "localization" come from? Why is the equivalence relation between the ordered pairs $(m,u),(m',u')$ with $ ...
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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 ...
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 ...
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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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
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 ...
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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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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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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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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?
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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
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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0answers
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 ...
2
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1answer
89 views

Cover of a direct summand

Let $L,N$ be $R$-modules. If $L$ and $L \oplus N$ have projective covers, is it true that $N$ admits a projective cover?
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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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1answer
42 views

dual algebra of a coalgebra

Given a coalgebra $A$ over a field $F$ (for example, $H_*(X:F)$, i.e. the homology of a space $X$ equipped with $\Delta_*$, where $\Delta: X\to X\times X$, $x\to (x,x)$) how to obtain its dual $A^*$ ...
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1answer
25 views

Defining the dual-comodule of a comodule?

As is well known, every left module has a dual, which is a right module. How does this work for comodules? More explicitly, does there exist a notion of the ...
3
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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 ...
2
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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 ...
1
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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 ...
3
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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 ...
4
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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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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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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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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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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 ...
3
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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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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 ...
3
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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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2answers
475 views

Isomorphism of modules + tensor product

Is it true that: $$M{\otimes}_{A}(A/I) \cong M/IM$$ and $$IM \cong I {\otimes}_AM$$ where $A$ is a commutative ring, $M$ an $A$-module, and $I \subset A$ an ideal.
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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 ...
2
votes
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 ...