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

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1answer
34 views

Chain map with one-sided inverse between isomorphic chain complexes quasi

Suppose that $C_\bullet$ and $D_\bullet$ are finitely generated isomorphic chain complexes. Let $f:C_\bullet\rightarrow D_\bullet$ a chain map with a one-sided inverse. Is it true that f is a quasi-...
4
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0answers
45 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 $B$-...
1
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0answers
17 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
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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 $S^{-1}A$-...
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2answers
53 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
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1answer
25 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 \frac{P}{N \...
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1answer
56 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
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1answer
29 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
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1answer
20 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
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1answer
18 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 ...
0
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1answer
82 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 M\...
2
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2answers
28 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 ...
2
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1answer
25 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 ...
0
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1answer
36 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
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1answer
61 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 $\chi$...
0
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1answer
23 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 ...
3
votes
2answers
50 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 ...
4
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1answer
23 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 ...
0
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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 $M$ ...
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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 ...
4
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2answers
84 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 $\...
3
votes
2answers
23 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 ...
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0answers
37 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 = \operatorname{...
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2answers
44 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
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0answers
55 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
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1answer
21 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 $\text{...
0
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3answers
43 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
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1answer
44 views

Relationship between modules and maximal ideals of a commutative ring [closed]

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
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1answer
26 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
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1answer
30 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
1answer
54 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
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1answer
37 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 $N=\text{Im}f\...
0
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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 $M=\ker f \oplus \text{Im}f$. $$$$ ...
0
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0answers
22 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
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1answer
34 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 $\mathbb{Z}$. $$$$ $$\...
2
votes
1answer
81 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 ...
3
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0answers
20 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 ...
0
votes
1answer
39 views

What's the importance of invariant factors?

I understand why the following theorem holds: If $R$ is a PID and $M$ is a finitely-generated torsion $R$-module, then there exist $q_1,...,q_s$, non-invertible elements of $R$, such that $q_i \...
0
votes
1answer
50 views

Each cyclic $R$-module is isomorphic to an $R$-module of the form $R/J$ [closed]

Let $R$ be a commutative ring with unit. I want to show that each cyclic $R$-module is isomorphic to an $R$-module of the form $R/J$ where $J$ is an ideal of $R$. $$$$ Could you give me some ...
6
votes
3answers
323 views

Why is (one of) the criteria for a Noetherian module not implied by Zorn's lemma?

One of the equivalent definitions of a Noetherian $R$-module is Every nonempty collection of $R$-submodules has a maximal element under inclusion so, what I'm wondering is, why is this not an ...
2
votes
0answers
19 views

Compatibility of Yetter-Drinfeld modules.

Let $H$ be a Hopf algebra. A Yetter-Drinfeld module over $H$ is a triple $(V, \cdot, \delta)$, where $\cdot : H \otimes V \to V$ , $\delta : V \to H \otimes V$ are actions and coactions respectively, $...
0
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2answers
20 views

Let $S$ be a subring of a commutative Noetherian ring $R$. Then $R$ is finitely generated as an $S$-module? [closed]

Let $S$ be a subring of a commutative Noetherian ring $R$. Then how can I show that $R$ is finitely generated as an $S$-module?
3
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0answers
20 views

Existence of alternative basis element in free module over a PID

first question on stack exchange, please let me know if I have made any errors with formatting or in general! :) Let $f_1,f_2, ...,f_s$ be a basis of a free module $V$ over a PID $R$. Suppose that $f=...
3
votes
2answers
66 views

For $R$-modules M, is $M\cong R^{\oplus n}\otimes_RM\cong M^{\oplus n}$?

I wanted to explicitly give a bilinear map $R^{\oplus n}\times M\longrightarrow M^{\oplus n}$ when trying to prove that $R^{\oplus n}\otimes_RM\cong M^{\oplus n}$ and ended up with $(r_1,...,r_n,m)\...
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0answers
27 views

Make a vector space into a module on ring of polynomials

I'm teaching myself about modules and I came across this exercise that I'm having trouble with (found here: http://www.math.uiuc.edu/~r-ash/Algebra/Chapter4.pdf). The question asks: Let $T$ be a ...
1
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3answers
55 views

A conceptual question in ring theory?

What is the main(conceptual) difference between an ideal of a ring and a submodule over a ring?
0
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2answers
48 views

For all $R$-modules $N$, $R\otimes_R N\cong N$

Why is this so? This statement is from Aluffi's book Algebra: Chapter 0 and seemingly so trivial that it deserves no proof. So working with the universal property of tensor products, how exactly does ...
1
vote
1answer
16 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 ...
0
votes
0answers
10 views

The reduced norm map $\operatorname{Nrd}: K_1(A)\to K^\times$

Let $K_1(A)$ be the Grothendieck $K_1$-group of the category of finitely generated projective $A$-modules where $A$ is a central simple $K$-algebra. I'd be grateful if someone could tell me if the ...
0
votes
1answer
17 views

What is the reduced norm map?

This is a basic question about the reduced norm homomorphism. Let $A$ be a central simple $K$-algebra and $P$ a f.g. projective $A$-module. I know that $\operatorname{End}_A(P)$ is also a central ...