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

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35 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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0answers
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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15 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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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
84 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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0answers
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
73 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
31 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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31 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
43 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 ...
3
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1answer
42 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
23 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
27 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
54 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 ...
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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 ...
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0answers
41 views

Gorenstein ring and projective module

I am new to this topic and would appreciate little explanation. Def: A commutative, unital ring $A$ is a cubic ring if $A$ is a free $\mathbb{Z}$-module of rank $3$. Def : A cubic ring $A$ is ...
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29 views

Projective Resolutions of Finite Type.

Definition. An $R$-module $M$ is said to be of type $FP_n$ if there is a projective resolution $\mathscr{P}$ of $A$ with $P_i$ finitely generated for all $i\leq n$. If the modules $P_i$ are finitely ...
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1answer
71 views

Self-injective ring but not semisimple?

It is well-known that if $K$ is a field, then $K[x]/(f(x))$ is a self-injective ring for any polynomial $f(x)$ in $K[x]$. On the other hand, we know that a ring $R$ being semisimple is equivalent to ...
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2answers
28 views

Injectivity of Inflation Homomorphism $H^1(G/K,A^K)\rightarrow H^1(G,A)$

I want to prove that injectivity of $inflation\ homomorphism$ $$0\rightarrow H^1(G/K,A^K)\rightarrow H^1(G,A)$$ where $K$ is normal. Proof : (a) If $x\in A^K$ then $gH\cdot x:= gx$ This is ...
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0answers
55 views

If $R$ is a integral domain then $TM$ is torsion module

Let $R$ be an integral domain. An $R$-module $M$ is torsion if $TM = M$. Prove that $TM$ is torsion. I'm confused with this exercise, i need to prove that $TTM = TM$ but we know that $TM$ is a ...
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28 views

Is this map an epimorphism

If A-->B is an epimorphism and if Hom(M,A)is isomorphic to A and Hom(M,B) is isomorphic to B, does this imply that Hom(M,A)-->Hom(M,B) is an epimorphism under the Hom(M,-) map?(A,B,M are R-modules)