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

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2
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1answer
41 views

divisible modules over Dedekind Domains

L. Fuchs in one of his articles says that: "divisible modules over Dedekind Domains can be completely characterized by numerical invariants". Please introduce me to a source in this respect. I so ...
0
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0answers
26 views

Embedding a ring in a direct product

If an $R$-module $C$ is a homomorphic image of a direct sum $⊕M$, where $M$ is an $R$-module, and $R$ could be embedded in a direct product $ΠC$, could $R$ be embedded in a direct product $ΠM$?
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0answers
27 views

How to compute cyclic/dihedral powers of modules?

The $n$-th cyclic power of an $R$-module $M$ is $$T^n_C(M)=M^{\otimes n}/\langle m_1\!\otimes\!m_2\!\otimes\!\ldots\!\otimes\!m_n-m_2\!\otimes\!\ldots\!\otimes\!m_n\!\otimes\!m_1;\, ...
1
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0answers
19 views

Does faithfully flat descent work using restriction of scalars rather than extension?

Vistoli's notes on fibred categories and descent - http://homepage.sns.it/vistoli/descent.pdf - introduce (section 4.2.1) descent on modules over a commutative ring. The idea is as follows: ...
0
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1answer
31 views

What are the generalized statements of given statements?

Let $R$ be a ring and $M$ be an $R$-module. Thm1. If $R$ is a division ring, then $M$ is free Thm2. If $R$ is commutative, the rank of $M$ is unique. First of all, are these statements ...
0
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1answer
20 views

Cauchy sequence in a valuation ring

From Janusz's book algebraic number fields, chapter 2. Let $K$ be a complete field with respect to a non-Archimedean absolute value $|\cdot|$. Let $R$ be its valuation ring with maximal ideal ...
2
votes
1answer
24 views

What is the standard notation for the submodule generated by a set?

In group theory, the standard notation for the subgroup generated by a set is $<A>$. In ring theory, the standard notation for the ideal generated by a set is $<A>$ while there is no ...
1
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1answer
35 views

When is a pure tensor equal to $0$?

Let $R$ be a commutative unital ring and let $A$ and $B$ be $R$-modules. For any $b\!\in\!B$ the set $$\{a\!\in\!A;\, a\!\otimes\!b\!=\!0\text{ in }A\!\otimes_R\!B\}$$ is a submodule of $A$ that ...
2
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0answers
24 views

What is the standard definition of Torsion element?

Here are two different definitions of torsion element. Let $M$ be an $R$-module and $m\in M$. Wikipedia: $m$ is a torsion element iff there exists a nonzero regular element $r$ (i.e. Not a zero ...
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0answers
37 views

Intersection of modules is equal to product.

If $B$ is a commutative ring and let $\mathcal{Q}_1,\ldots,\mathcal{Q}_n$ ideals relative primes. Let $M$ be a $R$-module. I don't sure if this is true. Then $$(\mathcal{Q}_1\cap\cdots ...
1
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1answer
21 views

Invariant factor decomposition algorithm in Dummit & Foote.

I have a question about a particular step of the invariant factor decomposition algorithm in Dummit and Foote on page 480. My question is about step $2$. Here is the general set up and the first three ...
1
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1answer
32 views

Torsion module decomposition into cyclic factors.

I am trying to resolve some confusion about the decomposition of a torsion module into cyclic factors. Let $R$ be a PID. Let $M$ be a finitely generated torsion $R$-module. That is, $M=T(M)=\{m \in ...
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0answers
19 views

Is the direction of containment right?

If $N$ is a submodule of $M$, then for any $x$ such that $xM = 0,$ then given arbitrary $n ∈ N, n ∈ M, xn = 0.$ Then the collection of $x$'s that annihilate $M$ contain the collection of $x$'s that ...
3
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1answer
34 views

Prove that $R/I$ is free if and only if $I=0.$

Prove that $R/I$ is free if and only if $I=0.$ Is my following proof OK? Assume $R/I$ is a free $R$-module. Then $\mathrm{Ann}(R/I)=0.$ That means $\{x \in ...
1
vote
1answer
15 views

Socle of a ring $R$

It is well-known that for an idempotent $e\in R$, the right $R$-module $eR$ is simple faithful if and only if $Re$ is a simple faithful left $R$-module. Now, I want to prove that when $Re$ is ...
1
vote
2answers
37 views

Finding an exact sequence and proving that it is split exact

Let $f : M \to M$ be an endomorphism of a module $M$ such that $f^2=f$.Then prove that, $M = \text{Im}\ f \oplus \ker f$. I was thinking that this can probably be proved if I can find an exact ...
1
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0answers
33 views

Endomorphism ring of semisimple modules

Can someone help me to prove the following statement, or introduce some reference about it? Statement: Suppose that $M$ is a semisimple $R$-module of finite type. There are division rings $D_1, ... ...
1
vote
1answer
75 views

Two modules not isomorphic if different number of elements?

Quick question: With $R=\mathbb{Z}, M=\mathbb{Z}, P=2\mathbb{Z}, Q=3\mathbb{Z}$ modules, can I conclude that $M/P = \mathbb{Z}_2 \ncong M/Q = \mathbb{Z}_3$ because they have a different number of ...
0
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1answer
56 views

Identifying the tensor product of two given modules over $\mathbb Z/2\mathbb Z$.

Suppose that $\pi$ denotes the finite ring of order $2$, $\mathbb Z_+$ the trivial $\pi$-module and $\mathbb Z_-$ the non-trivial $\pi$-module (i.e., the generator acts by multiplication by −1). I ...
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0answers
18 views

Question about showing that a module is a direct sum of modules of 3 given forms

Let π denote the finite ring of order 2, denote by $\mathbb Z_+$ the trivial π-module and by $\mathbb Z_−$ the non-trivial π-module (i.e., the generator acts by multiplication by −1). Suppose $A$ is a ...
1
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1answer
32 views

Artin-Wedderburn theorem and square dimension

Let $A$ be a finite-dimensional simple algebra over $\mathbb{C}$ of dimension $n$. By Wedderburn's theorem, we have that $A$ is isomorphic to a matrix ring $M_r(\mathbb{C})$, which is of dimension ...
0
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2answers
52 views

Left and right R-modules

OK, this question may be induced by my highly commutative mind and the wee midnight hours. Why do we bother with right R-modules vs left R-modules in view of the fact that given, say, a left R-module ...
2
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0answers
38 views

indecomposable summand

Let $M$ be an $R$ module. Is this true for start a proof that we say "Let $S$ denote the indecomposable summands of $M$"? In fact, I want to know whether any module over a Dedekind domain (or a ...
2
votes
1answer
19 views

Module of indecomposables

I've encountered the following definition, which I'm struggling to get my head around: Let $(A, \mu, \eta, \epsilon)$ be an augmented algebra, with augmentation ideal $I(A) = \ker(\epsilon)$. We ...
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0answers
36 views

Is submodule of Hilbert module a Hilbert module?

Let $R$ be a commutative ring with identity and $M$ be an $R$-module. $M$ is a Hilbert module if every prime submodule $P$ of $M$ equals the intersection of all maximal submodules of $M$ that contain ...
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0answers
34 views

Module Theory: about the genesis of the concept

I'm studying the module theory. But, I really want to know the history of the genesis of the concept of R-module. On my years as a student, my Algebra teacher presenteed to me the concept as a simple ...
2
votes
1answer
53 views

Why is the algebraic closure of $\Bbb{Q}$ not a finitely generated $\Bbb{Q}$-module?

Let $\overline{\Bbb{Q}}$ be the algebraic closure of $\Bbb{Q}$. I am trying to show that $\overline{\Bbb{Q}}$ is not finitely generated as a $\Bbb{Q}$-module, however I do not know where to go with ...
-1
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2answers
39 views

Show that $M$ is not isomorphic to $M/N \oplus N$ in general [closed]

Consider a submodule $N$ of a module $M$. Get an example showing that $M$ is not isomorphic to $M/N \oplus N$ in general.
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0answers
31 views

Quotients of finitely generated module are direct summands implies module is semisimple

Let $R$ be a ring with unity, and let $M$ be a finitely generated $R$-module. I wish to show that if every quotient of $M$ is isomorphic to a direct summand of $M$, then $M$ is a semisimple ...
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0answers
24 views

Free $A$ module

Let $A$ be a complex algebra and $A^{op}$ denote opposite algebra with multiplication $a \cdot b:=ba$. Is it obvious that $(A \otimes A^{op}) \otimes A^{\otimes n}$ is a free $A \otimes A^{op}$ ...
2
votes
0answers
11 views

First order differential calculus over algebra $A$

Let $A$ be a unital algebra and $d_u$ is universal differential given by $d_u(x)=1 \otimes x-x \otimes 1$. It is viewed as the map $A \to \Omega_u^1(A)$ where $\Omega_u^1(A)=\ker m$ is the kernel of ...
2
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0answers
25 views

Another construction of Specht module.

Let $\lambda$ be a partition of $n$, and $\lambda^*$ the dual partition (i.e. having the transposed Young diagram). Let $z_i$ be vectors in $\mathbb{C}^{\lambda_i^*}$, and$$F_\lambda = \prod_i ...
0
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0answers
22 views

Another question about free modules of finite rank over a PID

I am now trying to prove the following: Let $R$ be a PID and let $M$ be a free $R$-module of finite rank. If $N$ is a submodule of $M$ and $M/N$ is finite, then rank$(N)$ = rank$(M).$ Attempt at a ...
4
votes
1answer
74 views

$\varinjlim\operatorname{Hom}_R(N,M_i) = \operatorname{Hom}_R(N, \varinjlim M_i)$

Show that $\varinjlim \operatorname{Hom}_R(N,M_i) = \operatorname{Hom}_R(N, \varinjlim M_i)$ is true when $N$ is finitely generated and $R$ is noetherian. Do you think the noetherian condition is ...
0
votes
1answer
21 views

M an R-module, where M is a commutative ring, if $M≅R/I$ for some ideal I of R, then $M$ is a cyclic R-module

M is an R-module, where M is a commutative ring, if $M≅R/I$ for some ideal I of R, then show $M$ is a cyclic R-module. Note: $M$ is a cyclic R-module means $M=<m>$ for some $m\in M$. Please ...
2
votes
2answers
34 views

Show $(⊕M_α)/I(⊕M_α)\simeq ⊕(M_α/IM_α)$ for ${M_α}$ $R$-modules, and $I$ is an ideal of the commutative ring $R$

Let $(M_α)$ be a collection of $R$-modules, and $I$ is an ideal of the commutative ring $R$. Show that $(⊕M_α)/I(⊕M_α)$ is isomorphic to $⊕(M_α/IM_α)$ as $R/I$-modules. Please help, thanks a lot!
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0answers
24 views

how to define the map between the isomophism $(⊕Mα)/I(⊕Mα)\simeq ⊕(Mα/IMα)$?

May I ask how to define the map between the isomorphism $(⊕M_α)/I(⊕M_α)\simeq ⊕(M_α/IM_α)$?. where ${M_α}$ is a collection of $R$-modules, and $I$ is an ideal of $R$. we know $(⊕M_α)/I(⊕M_α)\simeq ...
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0answers
34 views

If $R$ is a commutative ring then prove that every left $R$-module is also a right $R$-module

$R$ is a commutative ring and $M$ is a left $R$ module. Define $\cdot : M*R \to M$ as $(x,a) \to x\cdot a$ by, $$x\cdot a=ax$$ To prove : $M$ is a right $R$ module. Proof: Let $x,y \in M$ and ...
3
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1answer
27 views

A left ideal need not be a right $R$ module

Let $R$ be a ring, it naturally follows that every left ideal of $R$ will also be a left $R$ module and an analogous result will also hold for right ideals of $R$. I was trying to show that a left ...
0
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1answer
52 views

Questions about subrepresentations of a representation of a quiver.

Let $Q$ be the quiver $\cdot \to \cdot \to \cdot$. Then $$ \mathbb{C} \to^{f} \mathbb{C} \to^g \mathbb{C} \quad (1) \\ 0 \to^{0} \mathbb{C} \to^0 0 \quad (2) $$ are two representations of $Q$, ...
0
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2answers
35 views

Baer's criterion, without unitary

Baer's Criterion says: A left R-module Q is injective if and only if any homomorphism g : I → Q defined on a left ideal I of R can be extended to all of R. In Hungerford's algebra, the R-module ...
2
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1answer
71 views

Is tilting theory extended also to arbitrary derived categories?

I was reading papers by Rickard ("Morita theory for derived categories") and Keller ("Derived categories and tilting") on tilting theory in derived categories, they seem to focus mostly on module ...
0
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1answer
36 views

Who to solve this linear modular equation system?

I have this equation system: a + b + c (mod 11) = 8 9a + 3b + c (mod 11) = 2 16a + 4b + c (mod 11) = 9 Unfortunately I totally don't know how to solve it. It is in general part of Lagrange's ...
0
votes
2answers
64 views

Isomorphism of two $\operatorname{Hom}$ modules

Let $R$ be a ring (associative, commutative, with unity) and $I\subset R$ is an ideal. Let $M$ be an $R/I$-module and $N$ an $R$-module. Is it true that $$\operatorname{Hom}_R(M,N)\cong ...
2
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1answer
55 views

Characterization of projective and injective modules

In Theorems 4.7 and 8.4 Hilton & Stammbach give two lists of 5 different characterizations of projective and injective modules, respectively. Even though I can follow the proofs they give, I'd ...
7
votes
4answers
218 views

Trouble understanding Eisenbud Exercise 2.19a

I'm working through the "Commutative algebra with a view toward algebraic geometry" book and stumbled onto an exercise I'm struggling to answer. Let $R$ be a ring and let $M$ be an $R$-module. ...
2
votes
1answer
34 views

Show that $R = R\cdot e \oplus Ann R(e)$

I have the following problem. Let $R$ be a ring and let $e$ be an idempotent in $R$. We consider $R$ as an $R$-module. Let $R \cdot e$ be the submodule generated by $e$ (since it happens in the ...
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votes
1answer
35 views

Show that the defined module is not finitely generated, but can be finitely generated with a modification

Define $R[U^{-1}]=\{\frac{r}{u}|r\in R, u\in U\}$, which is an R-module, where $R$ is an integral domain. In my example if $U=\{s^i|i\in\mathbb{Z}_{>0}\}$ for some $s\in R$, then prove that ...
0
votes
2answers
35 views

“Multiplying” by denominator of quotient module

Let $R$ be a ring, $A$ be an $R$-module, and $B, C$ be submodules such that $A = B \oplus C$. Then we can say that $\dfrac{A}{B} = \dfrac{B \oplus C}{B} \cong \dfrac{C}{B \cap C} = \dfrac{C}{0} \cong ...
3
votes
1answer
65 views

Is there a surjective homomorphism from $\oplus_{i=0}^\infty $ $\Bbb Z_p$ into $\prod_{i=0}^\infty$ $\Bbb Z_p$?

Is there a surjective homomorphism from $\oplus_{i=0}^\infty $ $\Bbb Z_p$ into $\prod_{i=0}^\infty$ $\Bbb Z_p$? Can this be shown by the order of element? Can we say this homomorphism is one-one, ...