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

learn more… | top users | synonyms

4
votes
0answers
36 views

How to recover a $K$-algebra from endomorphism algebra of forgetful functor?

I'm trying to work out how a finite-dimensional $K$-algebra $B$ can be recovered from the algebra of endomorphisms of the forgetful functor $\omega$ from the category of finitely generated left ...
1
vote
0answers
16 views

Proving that an element of a ring annihilates a module [duplicate]

Let $R$ be a commutative ring with $1$, $M$ be a finitely generated $R$-module, $\mathfrak{i}$ an ideal of $R$, and $\phi$ an $R$-homomorphism such that: $$1.\;\phi(M)\subseteq \mathfrak{i}M=M$$ i. ...
0
votes
2answers
50 views

How can I find a linearly independent subset of a set of vectors in $\mathbb{Z}^n$?

I have a set of $M$ vectors in the module $\mathbb{Z}^n$ ($M>n$) over $\mathbb{Z}$. Question 1: How can I find a linearly independent subset of these vectors? (so that others can be written as a ...
0
votes
1answer
10 views

$A$: ring, $S$ multi closed subset. $M:A$-module $\iota:M\to S^{-1}M, m\mapsto m/1$. $N'$: $S^{-1}A$-submodule, $N=\iota^{-1}N'$. Then $S^{-1}N=N'$

Let $A$ be a ring and $S\subset A$ be a multiplicatively closed subset. Let $M$ be an $A$-module and let $\iota:M\to S^{-1}M$ be the natural map given by $m\mapsto m/1$. let $N'$ be a ...
0
votes
1answer
22 views

Suppose $P, N$ are sub-modules of a module $M$. Then there exist natural isomorphisms..

Suppose $P, N$ are sub-modules of a module $M$. Then there exist natural isomorphisms $$(1) \ \ \frac{P+N}{P} \approx \frac{N}{N \cap P} \ \ \text{ and } \ (2)\ \ \frac{P+N}{N} \approx ...
0
votes
1answer
38 views

Show that these are the only $\mathbb{R}[x]$-submodule of $\mathbb{R}^2$ [on hold]

Let $T$ be the linear mapping from $\mathbb{R}^2$ to $\mathbb{R}^2$ that is given by a clockwise rotation of $\frac{\pi}{2}$. $T$ makes $\mathbb{R}^2$ an $\mathbb{R}[x]$-module. I want to show ...
2
votes
1answer
53 views

Solve $\begin{cases}x\equiv 1\pmod{5}\\x\equiv0\pmod{66}\\x\equiv6\pmod 7\end{cases}$

Solve $$\begin{cases} x\equiv 1\pmod{5}\,\,\,\qquad\qquad.1\\ x\equiv0\pmod{66}\qquad\qquad.2\\ x\equiv6\pmod 7\,\,\,\qquad\qquad.3 \end{cases}$$ My attempt: $\gcd(66,5,7)=1$ so I can apply the ...
1
vote
1answer
24 views

Free modules and their tensor product

can tensor product of two free non zero module over commutative ring with unity be zero? And can tensor product of two non zero vector spaces be zero space?
2
votes
1answer
15 views

If $M$ is a simple $R$-module, and an $F$-space, why does $End_F(M)\cong M^{\oplus\dim_F(M)}$?

Suppose a ring $R$ is an $F$-algebra for $F$ a field, and $M$ is a simple $R$-module and a finite dimensional $F$-vector space. We can endow $\operatorname{End}_F(M)$ with an $R$-module structure by ...
2
votes
1answer
18 views

Not free as a bimodule.

Let $R$ be a ring with 1. I am not following why the ring $R$ is free as a right or left module over itself but not as an $R$-bimodule. Clearly for any $r \in R$, $r=1r1$, so one is a basis as a ...
2
votes
2answers
26 views

Module of finite length $\implies$ finite direct sum of indecomposable modules

I'm trying to solve the following question. Let $M$ be an $R$-module of finite length (i.e, both Artinian and Noetherian). Prove that it is isomorphic to a finite direct sum of indecomposable ...
0
votes
1answer
74 views

In $A$-Mod, $M\oplus A\cong A\oplus A$ implies $M\cong A$

(Exercise from an introductory course in homological algebra) Whenever $A$ is a commutative ring with unit and $M$ an $A$-module, the following holds: $$M\oplus A\cong A\oplus A \Rightarrow ...
8
votes
1answer
181 views

Is $R/N(R)$ a faithfully flat $R$-module?

I'm studying recently faithfully flat modules and I'd like to know the following: Is $R/N$ faithfully flat as $R$-module, where $R$ is a commutative ring with unit and $N$ is the ideal of ...
6
votes
2answers
316 views

$M\oplus A \cong A\oplus A$ implies $M\cong A$?

Let $A$ be a commutative unital ring and $M$ an $A$-module. Suppose that $M\oplus A \cong A\oplus A$. Then is $M\cong A$? We have that both $M\oplus A$ and $A\oplus A$ are biproduct for $(A, A)$ ...
1
vote
0answers
50 views

Simple modules and homological algebra

Let $A$ be a $k$-algebra, and $M$ an $A$-module. If $\mathrm{Ext}_A^{1}(M,S)=0$ for every simple $A$-module $S$, then $M$ is projective. I know that this is true if $A$ is finite-dimensional, but if ...
1
vote
1answer
49 views

Natural action of $\mathbb{Z}G$ on $\mathbb{Z}$?

I'm studying projective modules and I'm having problem coming up with (or understanding) examples of non-free projective modules. I got that when a ring is a direct sum $R = A \oplus B$, both $A$ and ...
0
votes
1answer
21 views

A possible characterization of divisible modules

According to mathworld: Definition. Let $R$ denote a commutative ring and $M$ denote a module over $R$. Then $M$ is divisible iff for every $a \in R$, if $a$ is not a zero-divisor, then for all $x ...
2
votes
1answer
23 views

General procedure to prove something is a tensor product of modules

I'm trying to understand some proofs of statements of the form: Show that some module is the tensor product of two other modules. When I'm looking at these proofs I always see that they start ...
1
vote
0answers
30 views

Hopfian and Co-Hopfian Modules.

Let $M$ be a $R-$module. We say that: $M$ is a Hopfian module, if every epimorphism of $M$ is a monomorphism. $M$ is a Co-Hopfian module, if every monomorphism of $M$ is an epimorphism. Why do we ...
0
votes
1answer
27 views

Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ a homomorphism…

I need some help with this problem, it seems easy, but I don't get it. Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ an homomorphism. Suppose that for each $A$-modules $M$ ...
4
votes
2answers
81 views

The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category ...
0
votes
1answer
29 views

Proving equivalent formulations of Jacobson radical

I should start by saying I found this post Equivalent definitions of the Jacobson Radical which is about the same two formulations of the Jacobson radical but it didn't really to answer my question. ...
3
votes
1answer
55 views

$\chi : \text{Hom}(M,A) \otimes \text{Hom}(A,N) \to \text{Hom}(M,N) $

Let $M,N$ be $A$-modules. Consider the map. $$\chi : \text{Hom}(M,A) \otimes \text{Hom}(A,N) \to \text{Hom}(M,N) $$ $$f \otimes g \mapsto g \circ f $$ If $M,N$ are projective I do know that ...
0
votes
0answers
38 views

A module such that Ext$^i(M,A)=0$ for all $i$. [closed]

Let $A$ be a ring. Does it exist a module such that Ext$^i(M,A)=0$ for all $i$?
4
votes
1answer
448 views

Tensor products and polynomials with coefficients in a module

This is exercise $6$ in chapter $2$ on modules from Atiyah's and Macdonald's book. Let $M$ be an $A$-module and let $M[x]$ be the set of all polynomials in $x$ with coefficients in $M$. Then ...
3
votes
2answers
45 views

Are these conditions sufficient for a $\mathbb{Z}$-module to be free?

There exist modules over the integers that, like $\mathbb{Q}$, manage to be torsion-free without being free. Ergo, its probably worth looking for conditions $P$ such that "$P$ + torsion-free" is ...
3
votes
1answer
16 views

Endomorphism ring of an irreducible comodule

Given a Hopf algebra over $\mathbb{C}$, and a irreducible comodule $V$, I can show that Mor$(V,V) \simeq \mathbb{C}$, where Mor$(V,V)$ is the space of comodule maps from $V$ to $V$. My proof uses that ...
1
vote
0answers
12 views

Let $S$ be a set. For each $i\in\mathbb{N}\cup\{0\}$ let $E_i$ the free module over $\mathbb{Z}$ generated by S^{i+1}$. Define…

I need some help with this problem, I really don't undertand how to start, thanks. Let $S$ be a set. For each $i\in\mathbb{N}\cup\{0\}$ let $E_i$ the free module over $\mathbb{Z}$ generated by the ...
1
vote
1answer
15 views

Relationship between idempotents in semisimple ring.

Let R be a semisimple ring with identity. If $e$ and $e'$ are idempotents in R such that $Re \simeq Re'$ then there exists $a \in R^\times$ such that $e' = aea^{-1}$ Attempt I know from a previous ...
1
vote
2answers
101 views

Which Short Exact Sequences Can I Extract From A Doubly Infinite Exact Sequence?

I know how if we have a short exact sequence of $R$ modules, $0 \rightarrow A_1 \rightarrow A_2 \rightarrow A_3 \rightarrow 0$ , we can deduce properties about the known modules from the unknown ...
1
vote
2answers
18 views

A question on “law of congruence of modulus”

I have a quick question about the law of congruence of modulus, which states "Let $a•b≡a•c \,(\mod m)$, where $a$ is not equivalent to $0$,$ \mod m$. We can cancel $a$ only when $a$ and $m$ are ...
1
vote
0answers
34 views

Minimal presentation of a tensor product of modules

Let $(R,m)$ be a local commutative noetherian ring. Suppose I have two modules $M$ and $N$ over $R$ given in terms of minimal presentations $$ M = \operatorname{coker}A, $$ and $$ N = ...
-2
votes
2answers
34 views

Are the statements about the free $R$-module correct? [closed]

Let $R$ be a commutative ring with unit. If $F$ is a free $R$-module with finite rank, does it hold that each set of its generators contains a basis and that each linearly independent set of ...
1
vote
0answers
48 views

Show that they are isomorphic

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. I want to show that $\text{Hom}_R(R,M)\cong M$ as $R$-modules and $\text{Hom}_R(R,R)\cong R$ as rings. $$$$ I have done the ...
0
votes
1answer
20 views

Does this stand for $\text{Hom}_R(R/I, R/I)$ ?

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. We have that $\text{Hom}_R(R,M)\cong M$ as $R$-modules and $\text{Hom}_R(R,R)\cong R$ as rings. Do we have then also that ...
0
votes
3answers
40 views

Is $Z/mZ\otimes Z \cong Z/mZ$?

I'm reading a Homological Algebra book that states this in some point without proving. I was trying to prove it and it seems to me that the first module is infinite and the second is not.
0
votes
1answer
37 views

Relationship between modules and maximal ideals of a commutative ring

Let $A$ be an integral domain, $M$ an $A$-module, and $m\in M$. Now for all maximal ideals $\mathfrak{m}$ there exists an $n\notin \mathfrak{m}$ such that $nm=0$. Why does this mean that $m=0$?
2
votes
1answer
19 views

If $I$ is finitely generated nilpotent and $R/I^{n-1}$ is noetherian then $R$ is noetherian

If $I$ is a finitely generated ideal of a commutative ring $R$ with $1$ such that $I^n = \{0\}$ and $R/I^{n-1}$ is noetherian, then $R$ is also noetherian. I don't know what I should do. If I can ...
0
votes
1answer
29 views

Kernel of the map given by $j(m)=1\otimes m$ is in torsion module

Let $R$ be an integral domain, $K$ its field of fractions and $M$ an $R$-module. I want to show that the kernel of the map $j:M\rightarrow K\otimes M, m\mapsto 1\otimes m$ is contained in the torsion ...
-1
votes
0answers
27 views

How can we conclude from that that $I$ is a principal ideal? [duplicate]

Let $R$ be a commutative ring with unit. I want to show that if $I$ is an ideal of $R$ then $I$ is a free $R$-module iff it is a principal ideal that is generated by an element $a$ that is not a ...
2
votes
1answer
69 views

Question regarding matrices with same image

Let $A$ and $B$ be $m\times n$ matrices over $\mathbb{Z}$ such that image($A$) = image($B$), where $A$ and $B$ are considered as maps $\mathbb{Z}^n \rightarrow \mathbb{Z}^m$. Does there exist an ...
6
votes
2answers
396 views

Some exact sequence of ideals and quotients

I saw an exact sequence of ideals $$0 \rightarrow I \cap J\rightarrow I \oplus J \rightarrow I + J \rightarrow 0$$In this sequence, maps are ring homomorphisms or module homomorphisms? And how the ...
4
votes
0answers
88 views

Testing if a submodule is free

This is hopefully a very simple question. In Gröbner Bases in Commutative Algebra by Ene and Herzog, I find the Problem 4.11, which says ($S$ here is a polynomial ring over a field $K$, $S=K[x_1\ldots ...
2
votes
2answers
447 views

Proving that the length of the sum and intersection of two finite length modules is finite.

I am trying to solve the following question: Let $M_1, M_2 \subset M$ be finite length submodules of a module $M$ (where $M$ need not be of finite length). Show that $ \ M_1 + M_2 = \{x_1 + x_2 \in M ...
0
votes
0answers
50 views

Show that it is equal to $\ker f \oplus \text{Im}f$ [duplicate]

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. I want to show that if $f:M\rightarrow M$ is a $R$-module homomorphism such that $f^2=f$ then $M=\ker f \oplus \text{Im}f$. $$$$ ...
-1
votes
1answer
49 views

The endomorphism ring is a field [closed]

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. I want to show that the endomorphism ring $\text{End}_R(M)=\text{Hom}_R(M,M)$ of a simple $R$-module is a field. $$$$ We have ...
1
vote
1answer
36 views

Show that it is equal to $\text{Im}f\oplus \ker g$

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. I want to show that if $f:M\rightarrow N$ and $g:N\rightarrow M$ are $R$-module homomorphisms such that $gf=1_M$ then ...
0
votes
0answers
13 views

Direct summand of module

Let $S$ be a commutative (associative) ring with $1$, which is an extension of a commutative ring $R$ (with same $1$). Suppose that $S$, as an $R$-module, is finitely generated projetive and faithful. ...
0
votes
1answer
33 views

Annihilator - Product of cyclic groups

Let $M$ be the abelian group, i.e., a $\mathbb{Z}$-module, $M=\mathbb{Z}_{24}\times\mathbb{Z}_{15}\times\mathbb{Z}_{50}$. I want to find the annihilator $\text{Ann}(M)$ in $\mathbb{Z}$. $$$$ ...
3
votes
0answers
17 views

$R=\{ m+nr\sqrt{2} \mid m,n \in \Bbb Z \}$ and $I_{a,b}=\{ ma+n(b+r\sqrt{2}) \mid m,n \in \Bbb Z \}$

Let $r$ be a natural number and $R=\{ m+nr\sqrt{2} \mid m,n \in \Bbb Z \}$. We can show that $R$ is a subring of the ring $\Bbb Q [\sqrt{2}]$. My questions are as follows: $(1)$ Suppose that a ...