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

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1answer
12 views

Why does $\langle 1/p^n+\mathbb{Z}\rangle\supseteq\langle 1/p^m+\mathbb{Z}\rangle$ imply $n\geq m$?

I'm trying to show that if $G=\{a/p^n\in\mathbb{Q}:a\in\mathbb{Z},n\geq0\}$ for a fixed prime $p$, then the quotient $G/\mathbb{Z}$ is Artinian. One minor detail I need to finish is $$ \langle ...
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2answers
68 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) ...
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0answers
17 views

Dual object for modules

I am trying to prove the following: $R$ a commutative ring, an $R$-module $M$ has a dual (in the categorical sense) if and only if $M$ is projective and finitely generated. Any hint on how to prove ...
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1answer
104 views

Morita contexts without tears

My question is: Has anybody seen Morita contexts introduced as it is done below? I first intended this as an answer to the question "Reference request: Morita contexts" by Bey, but then decided to ...
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1answer
40 views

The condition that a ring is a principal ideal domain

If $R$ is a nonzero commutative ring with identity and every submodule of every free $R$-module is free, then $R$ is a principal ideal domain. What I don't know is how to show that every ideal is ...
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1answer
46 views

Does $ax\in\mathfrak{m}I$ with $x\in I\setminus\mathfrak{m}I$ and $a \in R$ imply $a\in\mathfrak{m}$ for an invertible fractional $R$-ideal $I$?

Let $R$ be an integral domain, $\mathfrak{m}$ a maximal ideal of $R$, and $I$ an invertible fractional $R$-ideal. If $x \in I \setminus \mathfrak{m}I$ and $a \not\in \mathfrak{m}$, do we have $ax ...
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1answer
37 views

Rank of abelian groups

I have read that given a $\mathbb{Z}$-module $M$, the maximal number of $\mathbb{Z}$-linear independent elements is given by $\operatorname{rank}M=\dim_\mathbb{Q}(\mathbb{Q}\otimes_\mathbb{Z}M)$. ...
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36 views

Prove there are some elements in a commutative module [duplicate]

let R be a commutative ring and I is an ideal in R and also M is finite generating module on R. If $\varphi:\:M \to M$ be a homomorphism and $\varphi(M)\subset IM$ .Prove there are some elements ...
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1answer
29 views

Misunderstanding in Cartan-Eilenberg?

In Cartan Eilenberg's Homological algebra, page 13 it says: If $\Gamma$ is a principal ideal ring, then each ideal $I$ of $\Gamma$ is isomorphic with $\Gamma$, thus $I$ is free and $\Gamma$ is ...
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1answer
23 views

Projectiveness of direct product of projectives

It is well-known that a direct sum of modules is projective if and only if each summand is projective, and a direct product of modules is injective if and only if each of the modules are injective. ...
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49 views

Describe the automorphisms of this $\mathbf{Z}$-module

This follows up on what I thought was a good question which has now been deleted, asking about the automorphisms of the multiplicative group $\mathbf{Q}^{*}$. Is there a relatively simple ...
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1answer
29 views

counter example in $R$-modules law

Let $M$ be an $R$-module and $K$, $L$ and $N$ submodules. I would like to find a counterexample to the equality $$N\cap (K+L) =(N\cap K)+(N\cap L)$$ I can prove the equality is true when $K ...
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2answers
28 views

group of module homomorphisms is a module

I am trying to solve the following problem: Let $M$ and $N$ be two left $A$-modules. Prove that $Hom_A(M,N)$ has a left $Z(A)$-module structure with: $(a.f)(m)=a.f(m)$. Show $Hom_A(A,N) \cong N$ as ...
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0answers
16 views

Modules that allow an infinite exact sequence on them

I'm looking for a characterization of modules that fit the following property: (*) There's an infinite exact sequence with $\phi_i \neq 0 \space \forall i\in I $ $\require{AMScd}$ \begin{CD}\cdots ...
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0answers
24 views

A question about injection of a quotient in a direct summand

Let $M$ be a nonsingular right $R$-module in the sense that $Z(M)=\{m\in M : \operatorname{ann}(m)\text{ is essential in }R_R\}=0$. Could one find an $R$-monomorphism from the quotient $M/\bigcap N_i$ ...
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15 views

Goldie closure of the zero module

In "Lectures on Rings and Modules" by T.Y. Lam, it is claimed in Example (7.33) that the Goldie closure $cl(0)$ of the zero module $0$ on the ring $R=\left [\begin{array}\ \mathbb Z & \mathbb Z_2 ...
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1answer
85 views

exact sequence with infinite element

How to prove: Define exact sequence with infinite element from $R$-modules and $R$-homomorphism ,then exact sequence with infinite element from $\Bbb Z$-modules and $\Bbb Z$-homomorphism : ...
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1answer
64 views

Finite cyclic $\Bbb Z$-module and exact sequence

Suppose $M$ is $\Bbb Z$-module, cyclic, finite. How to prove $\require{AMScd}$ \begin{CD} 0 @>>> \Bbb Z @>>> \Bbb Z @>>> M @>>>0\\ \end{CD} is ...
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1answer
47 views

a problem in exact sequence

Suppose $\require{AMScd}$ \begin{CD} @.\acute M @>f>> M @>g>>\check M @>>> O\\ @. @V \alpha V V\ @VV \beta V @VV \gamma V @. \\ O@>>> \acute N ...
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28 views

is there an epimorphism from X to A?

A is X-generated means that there exists an epimorphism from a direct sum of X to A). My question is: If A is a submodule of X and also X-generated,does this imply that we have an epimorphism from X ...
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1answer
75 views

Finitely generated prime ideal and annihilator

Suppose $R$ is a commutative ring, $P$ is a prime ideal of $R$, $P$ is finitely generated, and $\operatorname{Ann}(P)=0$. Show that $$\operatorname{Ann}(P/P^2)=P.$$ These are my efforts: ...
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21 views

Module notation

Suppose I have a ring $R$ whose underlying group is $G$. Now suppose I read the definition: Let $M$ be an $n$-dimensional left module over $R$. Is this a conventional way of defining $M$ as ...
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1answer
46 views

A relation in a finitely generated module

Suppose $R$ is a commutative ring, $I$ is an ideal of $R$, and $M$ a finitely generated $R$-module s.t. $M=IM$. How to prove: $$\exists a \in I \text{ such that } (1-a)M=0. $$ I tried to solve: ...
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44 views

Rings and modules

Let $R$ be a ring in which every maximal ideal is a direct sum of cyclic $R$-modules. Now let $I$ be a proper ideal of $R$. What is the structure of $I$. Is it true that $I$ is a direct sum of cyclic ...
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1answer
111 views

A relation involving an endomorphism of a finitely generated module

Suppose $R$ is a commutative ring, $I$ is an ideal of $R$, and $M$ a finitely generated $R$-module. (Usually in this problem $R$ includes $1_R$.) Let $\phi : M \to M$ be an $R$-homomorphism, and ...
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1answer
115 views

Tensor Product, Exterior Power and Splitting

Let $M$ be a $\mathbb{Z}$-module and consider the submodule $K=\langle m\otimes m\mid m\in M\rangle$ of $M\otimes M$. Under what conditions does the SES $$0\to K\to M\otimes M\to M\wedge M\to 0$$ ...
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1answer
28 views

Submodule iff subgroup?

It is late at night and time for another silly question: Is it true that a subset $S$ of an $R$-module $M$ is a submodule if and only if it is a subgroup of $M$ as an abelian group? Of course, by ...
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1answer
40 views

Example of a ring $R$ such that $R\otimes_R R\not\simeq R$

I want to show the following statement: If a ring $R$ is commutative and $I,J\triangleleft R$, then $$ R/I\otimes_R R/J\simeq R/(I+J). $$ I can easily show this, assuming that $1\in R$. What can be a ...
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1answer
53 views

If $p \in \operatorname{Ass}M$, then $R/P \subset M$.

Let $R$ be a commutative ring with unity. $M$ an $R$-module. Then $P \in \operatorname{Ass}M$ if and only if there is a submodule $N\subset M$ such that $R/P \cong N$. ...
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1answer
32 views

“Adjugate” of an endomorphism of a finite-rank free module

If $M$ is a free module of finite rank $n$ over a commutative unitary ring and $a$ is an endomorphism of $M$, consider the endomorphism $\hat a$ of $M$ defined by the identity $$ x_1\wedge ...
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2answers
49 views

A question on endomorphisms of an injective module

This is a homework question I am to solve from TY Lam's book Lectures on Modules and Rings, Section 3, exercise 23. Let $I$ be an injective right $R$-module where $R$ is some ring. Let $H= ...
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25 views

How to solve equations with mod?

How to solve this: $18^x\mod6031=4273$ ? Let $18^x = t$, then: $\ t\mod6031=4273,\ t = 6031n+4273$. But then I can't make the reverse exchange. P.S. By bruting I got the following roots: $65,\ ...
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2answers
39 views

Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module.

Exercise: Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module. I'm not sure about all different kind of modules, but this is a question of a book about ...
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1answer
23 views

Any artinian chain ring is self-injective.

Let $R$ be an artinian chain (uniserial) ring. I want to prove that $R$ is self-injective, that is, any $\alpha:I\rightarrow R$ right $R$-module homomorphism can be extended to a homomorphism ...
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1answer
12 views

Making a module a vector space

Let $G$ be an abelian group whose elements are all of prime order $p$. Naturally, $G$ is a $\Bbb Z$-module. Now take a maximal ideal $(p)$ and divide the scalar ring with it. Then one can consider ...
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1answer
33 views

Show the module of continuous functions of antiperiod $\pi$ is not cyclic.

Homework Hint Needed: Let $R$ be the ring of all continuous functions on $\mathbb{R}$ of period $\pi$. Let $S$ be the $R$-module of all continuous functions on $\mathbb{R}$ of antiperiod $\pi$. So ...
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1answer
26 views

The definition of $A'[\phi]$

I'm having trouble understanding the following definition from my textbook: If $M$ is an $A$-module and $\phi: M \longrightarrow M$ an $A$-linear endomorphism of $M$, I write $A'[\phi] \subset$ ...
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32 views

Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$.

Let $A\in\operatorname{M}_n(F)$. How to prove the following identity. $$\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}.$$ Here $m_A(x)$ the minimal polynomial of $A$ and $C(A)$ is the ...
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0answers
19 views

Question on module theory (Flat module)

My question is: - Can a flat module over commutative ring be a torsion module? Thank you!
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1answer
44 views

Example of a commutative square without a map between antidiagonal objects?

In an abelian category, can there be a commutative diagram of (vertical/horizontal) exact sequences $$ \require{AMScd} \begin{CD} 0 @>>> N @>>> M\\ @. @VVV @VVV \\ 0 @>>> X ...
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1answer
23 views

Showing that an $(S,T)$-bimodule is a right $S^{op} \otimes_\mathbb{Z} T$-module

The questions I'm trying to answer is as follows (all rings are unital): Let $_{S}B_{T}$ be an $(S,T)$-bimodule, and let $R = S^{op} \otimes_\mathbb{Z} T$. Show that $B$ can be made into a right ...
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1answer
43 views

Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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0answers
32 views

Endomorphism rings of a $k$-algebra

Let $k$ be a field and $R$ be the $3$-dimensional $k$-algebra $\left [\begin{array}\ k & k \\ 0 & k \end{array} \right ]$. I have two conjectures: (1) Is any simple right $R$-module has ...
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1answer
58 views

Composite function of $R$-module

Let $M$, $N$, $P$ be $R$-module and functions $f:M\rightarrow N$, $g:M\rightarrow N$, $h:N\rightarrow P$, $k:P\rightarrow M$, which need not be homomorphisms. Define $f+g:M\rightarrow N$ by ...
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0answers
24 views

Confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky

I am confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky (Chapter 10, page 441). Which is as follows: Let $A=\oplus_{g \in G} A_g$ be a $G$-graded ...
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1answer
28 views

$H^1(G,A)\rightarrow H^1(H,A)$ is onto

Note that $H^1(G,A)\rightarrow H^1(H,A)^{G/H}$ where $A$ is $G$-module and $H$ is a subgroup in $G$ But I suspect that ${\rm Res}\ : \ H^1(G,A)\rightarrow H^1(H,A)$ may be onto since $ f\in C^n ...
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1answer
55 views

Why is axiom of choice needed? (Equivalent conditions for Noetherian)

Let $R$ be a commutative ring with $1$ and let $M$ be a left $R$-module. On page 458 of Dummit and Foote's Algebra, 3rd edition, they show that $M$ is Noetherian (i.e. satisfies A.C.C. on submodules) ...
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1answer
30 views

Proving a submodule is Noetherian

I'm doing some independent study with a professor in Ring/Field theory (I'm an undergraduate) and I have been having a hell of a time wrapping my head around problems involving Noetherian rings and ...
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1answer
43 views

Matrices over PID

Let $R$ be a PID and $A,B\in\operatorname{M}_n(R)$ are $n\times n$ matrices such that $\det(A)\sim\det(B)\neq0$,i.e., the ideals generated by $\det(A)$ and $\det(B)$ are the same, does there exist ...
1
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1answer
53 views

A question about projective modules.

Suppose that we have a commutative ring $R$ with an idempotent $e$, and $M$ an $R$-module such that $Me$ is $Re$-projective. I am interested to know under which conditions this implies that $M$ is ...