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

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2answers
19 views

Cosets of submodule

Suppose we have a ring $R$ and an ideal $J \subseteq R$. Let $M$ be an $R$-module and $N$ be a submodule of $M$. Assume for two elements $m$ and $m'$ of $M$ we have $$m+ JM=m'+JM.$$ Since $N ...
3
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0answers
24 views

Elementary divisors for chains of submodules

Given free modules $N \le M$ of finite rank over a PID $R$, it's well-known that there is a basis $\{x_1,\ldots,x_n\}$ of $M$ and there are $e_1,\ldots,e_n \in R$ such that $\{e_ix_i\mid e_i \neq ...
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1answer
20 views

A $R$-module over a ring $R$ with identity is free iff it is a direct sum of copies of $R_R$, where $R_R$ is $R$ considered as a module over itself

Let $R$ be a ring with identity and $M$ be an $R$-right module. Then $M$ is called free over $X \subseteq M$ if for every module $N$ with mapping $\alpha : X \to N$ we can extend it uniquely to a ...
0
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2answers
28 views

Generators for a free submodule of a free module

Suppose we are given a finitely generated free module $M$ over a ring $R$. Assume $N \subseteq M$ is a free submodule of $M$. If $B$ is a basis for $M$, does it follow that there exists a subset $A ...
5
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3answers
134 views

nilpotent endomorphism on finitely generated modules over a domain

If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$. The cases $ m \le n$ is trivial. When $m>n$, I don't have much ...
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0answers
15 views

Uniqueness submodule of cyclic module

I'm having some problems with an exercise from Hungerford's Book of Algebra. It states: Let $R$ be a PID, and $A$ a unitary $R$-module such that $A$ is cyclic of order $r$. a) Prove that ...
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2answers
25 views

CG-modules: what does this notation mean?

I am trying to solve a question, but I do not know what the notation used means. If anyone could help me out that'd be great! I don't need help doing the proof, just what the notation means would be ...
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0answers
28 views

If $f : M\otimes_A A/m \to N\otimes_A A/m$ is surjective , so is $f : M \to N$. [on hold]

Let $A$ be a local ring with maximal ideal $m$. Let $f : M \to N$ be a morphism of $A$-modules, where $N$ is finitely generated. Show that if the map $f : M\otimes_A A/m \to N\otimes_A A/m,\quad ...
1
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1answer
12 views

Kernel of $M\to M[U^{-1}]$ and primary decomposition of $(0)$

I am working on exercise 3.12 from Eisenbud's Commutative Algebra and I am having trouble parsing the question. Let $M$ be a finitely generated module over the Noetherian ring $R$. Given any ...
0
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1answer
56 views

$\mathbb C$-dimension of vector space $\mathbb C\otimes_{\mathbb R}\mathbb C$

Let $\mathbb R$ be the field of real numbers, $\mathbb C$ be the field of complex numbers. Consider $\mathbb C\otimes_{\mathbb R}\mathbb C$ as a $\mathbb C$-vector space via $a(b\otimes c) := ab ...
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0answers
34 views

Matrices representing a map between free modules of infinite rank and Fitting's Lemma (Eisenbud)

p.497 of Commutative Algebra with a View Toward Algebraic Geometry, Eisenbud: If $\phi: F \rightarrow G$ is a map of free modules, then $I_j\phi$ is the image of the map $$\Lambda^j F ...
3
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2answers
151 views

Example of a Tensor Product of Modules with Non-Decomposable Elements

Given a ring $R$ and $R$-modules $A_R$ and $_{R}B$, we define the tensor product $A \otimes_R B$ as the free abelian group on $A \times B$ modded out by the subgroup generated by the elements of ...
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2answers
28 views

what is the difference between finitely generated module and finitely generated free module?

I am still confused about the difference between free module and finitely generated module. For example, $Z/2Z$ is finitely generated module, but why it is not finitely generated free module? What is ...
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1answer
10 views

If $M/N$ and $N$ are noetherian $R$-modules then so is $M$

Let $M$ be an $R$-module. I want to show only using the definition of noetherian that if $N$ is a noetherian submodule of $M$ such that $M/N$ is noetherian, then $M$ is noetherian. I know that if $M_1 ...
0
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1answer
28 views

Ideals and submodules are the same [on hold]

My teacher has told me that for an R-module, that I is an ideal of R if and only if I is an R-submodule of R. I know this is true but I was wondering why? IS there an official proof?
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0answers
33 views

Every non-Noetherian module has a submodule maximal with respect to being not finitely generated. [duplicate]

Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated whenever $N<A\leq M$. The question is related to If $M$ isn't ...
0
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1answer
24 views

Structure of induced module for subgroup $H \le G$, confused by mixing up notions of $FH$-submodule and $F$-subspace

The following is a question on a passage in the book A Course in the Theory of Groups by Derek Robinson. There he constructs the induced module $M^G$ from a $FH$-module for some subgroup $H \le G$ of ...
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1answer
18 views

Computing generators for a finitely generated module

I came across this problem yesterday: Let $R$ be a ring and $M$ an $R-$module. $\varphi:R^n\to M$ is a surjective $R-$module homomorphism if and only if $M$ is finitely generated. Given the set of ...
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1answer
73 views

When is an ideal also a ring, and could then be anything said about its relation to the original ring

If $R$ is a ring with unity $1$, then $S \subseteq R$ is called a subring if it is itself a ring with $1 \in S$. A subset $I \subseteq R$ is called an ideal if it is a group with respect to addition ...
3
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1answer
39 views

In proof of $|G| = n_1^2 + n_2^2 + \ldots + n_h^2$ how could equality of dimensions concluded from ring isomorphism

In Derek Robinson, A Course in the Theory of Groups on page 224 he proves: Let $G$ be a finite group and let $F$ be an algebraically closed field whose characteristic does not divide the order of ...
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2answers
51 views

motivation for the direct limit [closed]

I know just the very basics on Category Theory and that's why I'm going to ask a stupid question. I'm trying to get an intuition for direct limits for my course on Commutative Algebra. All the books ...
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0answers
15 views

Show that all simple modules of ${M_n}{(D)}$ are isomorphic to $D^n$, where $D$ is a division ring. [duplicate]

I am trying to solve part (c) of the following Representation Theory question: Let $D$ be a division ring and let $n$ be a positive integer. For $ 1 \leq l \leq n $ let $$C_l= \{A = (a_{ij}) \in ...
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1answer
20 views

Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$, where $N_G = {\sum}_{g \in G} {g} \in F[G]$. [duplicate]

I am trying to solve this Representation Theory question: Let $F$ be a field and $G$ a finite group. Let $N_G = {\sum}_{g \in G} {g} \in F[G]$. Show that the left ideal $(N_G) \subset F[G]$ is a ...
1
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1answer
23 views

Show ring isomorphisms $End_R{({S_1}^{n_1} \oplus \ldots \oplus {S_r}^{n_r} )} \cong End_R{({S_1}^{n_1})} \times \ldots \times End_R{({S_r}^{n_r})}$

I have been struggling with this Representation Theory question for the past week: Let $R$ be a ring and $S_1, \ldots, S_r$ simple $R$-modules with $S_i$ not isomorphic to $S_j$ whenever $i \neq j$ ...
0
votes
1answer
24 views

Exact Sequences of Modules and Rank

Suppose that: $0 \rightarrow M_{1} \rightarrow M \rightarrow M_{2} \rightarrow 0$ Is an exact sequence of $R$-modules (for $R$ commutative integral domain). Show that $\mathrm{rk}(M) = ...
1
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1answer
22 views

Poincare series and the Hilbert polynomial of $A = A_0[X_1,\dots , X_s]$

Let $A = A_0[X_1, \dots , X_s]$ be a polynomial ring in $s$ variables over an Artin ring $A_0$. This is a graded ring, and can be regarded as a graded module over itself. 1. What are the ...
0
votes
1answer
44 views

Poincare series and Hilbert polynomial of some graded modules [on hold]

Let $k$ be a field, and let $k[X, Y ]$ be the polynomial ring in two variables equipped with the usual grading such that $\deg(X) = \deg(Y ) = 1$. Consider the ideals $I = (X, Y^2)$ and $J = (X^2, ...
1
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1answer
51 views

Tensor product of Hom-module and another ring

Let $A$ be a local noetherian ring, $B$ and $C$ are finitely generated $A$-algebras and $M$ is a finitely generated $B$-module. Is the natural morphism $\mathrm{Hom}_B(M,B) \otimes_A C \to ...
1
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1answer
26 views

Submodule of a free module with the same rank is a direct summand

Let $M$ and $N$ be finite rank free modules over a ring with invariant basis number such that $N \subseteq M$ is a submodule of $M$. Suppose $M$ and $N$ have the same rank $n$. Is it possible for ...
0
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1answer
24 views

where do elements go under multiplication in a graded module?

Assume that $M = \bigoplus_{n = 0}^\infty M_n$ is a graded $A$-module, where $A = \bigoplus_{n = 0}^\infty A_n$ is a graded ring. We have by definition $A_m M_n \subset M_{m + n}$. Does this mean that ...
0
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0answers
17 views

Why is $(A \otimes_\mathbb{Z} V_p )\bigcap R(G) = V_p$

I am reading the proof of brauers theorem in Serres book Linear Representations of finite groups and I have trouble understanding Lemma 5(page 75). Let $G$ be a group of order $g$. First let $g=p^nl$ ...
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0answers
26 views

If F is a simple module, is $F^n $ semi simple?

If F is a simple module, is $F^n $ semi simple? I'm assuming this is true by the classification theorem
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0answers
34 views

Origin of the name “Cauchy sequence” in abstract algebra

I know the origin in analysis as it is derived from our good old chap Cauchy himself. However I have read about Cauchy sequences in algebra, I'll use groups for this one, let $G$ be a group and ...
0
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0answers
36 views

Two actions of rings in localization of modules

If $\mathfrak p,\mathfrak q$ are prime ideals of $R$. And $M$ is a $R$-module. Then $(M_\mathfrak p)_\mathfrak q$ is a $R_\mathfrak p$ and $R_\mathfrak q$-module under the action respectively 1) ...
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2answers
54 views

$\overline\phi: M/IM \to N/IN$ is surjective, then $\phi$ is surjective.

Let $I$ be a nilpotent ideal in a commutative ring $R$, let $M$ and $N$ be $R$-modules and let $\phi : M \to N$ be an $R$-module homomorphism. Show that if the induced map $\overline\phi: M/IM \to ...
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0answers
41 views

Some questions about S.Roman, “Advanced Linear Algebra”

Question for those who have studied Roman's book "Advanced Linear Algebra". How self-contained is this book. Can I study determinants directly from this in context of exterior algebra and tensor ...
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3answers
43 views

When does isomorphism between submodules induce an isomorphism between modules

Let $R$ be a ring and let $M$ and $N$ be $R$-modules with submodules $L$ and $P$, respectively. Assume we know $M \cong N$ as $R$-modules. Further, suppose we are given an isomorphism $$\psi: L \to ...
2
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1answer
50 views

Endomorphism of a finitely generated module and finite length of $\operatorname{Coker}f$

Let $M$ be a finitely generated module over a noetherian ring $R$ and suppose that $\dim R≤1$. And let be a one to one homomorphism $f:M\to M$. It is true that $\operatorname{Coker}f$ has finite ...
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2answers
49 views

Is $\mathbb{Q}$ artinian as a $\mathbb{Z}_{(2)}$-module?

Let $\mathbb{Z}_{(2)}= \{ \frac ab \in \mathbb{Q} : 2\not \mid b \}$ be the localization of $\mathbb{Z}$ at the prime ideal $(2)$. We can consider $\mathbb{Q}$ as a $\mathbb{Z}_{(2)}$-module by the ...
0
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1answer
55 views

Finite length module over Noetherian ring

Let $M$ be a finitely generated module over a noetherian ring $R$ and suppose that $\dim R\leq 1$. Then $M$ has finite length. A preliminary lemma: There exists an chain $\{0\}=M_0\subsetneq ...
2
votes
1answer
25 views

Help showing $H_n=\sum_{h \in H}h \in F[H]$ is a simple submodule

Given $F$ a field and $H$ a finite group and $H_n=\sum_{h \in H}h \in F[H]$, I'm trying to show that the left ideal $(H_n)\subset F[H]$ is a simple submodule. Now I'm not really sure what the ideal ...
0
votes
1answer
30 views

Show endomorphism $\phi$ is determined by $\phi(e_1)$

Say $\phi \in End_{M_n(D)}(D^n) $ I'm trying to show $\phi$ is determined by $\phi(e_1)$ and that $\phi(e_1)=de_1$ where $d \in D$ To show it's determined by $\phi(e_1)$ I have used the property that ...
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1answer
38 views

Trying to show $End_R(U_1 \oplus U_2) \cong End_R(U_1) \times End_R(U_2)$

I'm trying to prove $End_R(U_1 \oplus U_2) \cong End_R(U_1) \times End_R(U_2)$ Where $U_1, U_2$ are simple R-modules s.t $U_1 \not\simeq U_2$ So far I have constructed a mapping $\phi \mapsto (\pi_1 ...
0
votes
0answers
25 views

$R[S]\cong \mathbb{Z}[S]\otimes_{\mathbb{Z}} R$, $R\otimes_{\mathbb{Z}} \mathbb{Z}\cong R?$

I don't know much about tensor products in general but for for some lectures I need basics about tensor products of rings, modules and abelian groups. I have the following questions: 1)Let $R$ be a ...
0
votes
1answer
31 views

Let $A$ be a local ring, $M$ and $N$ finitely generated $A$-modules. Show that if $M \otimes N = 0$ then $M=0$ or $N=0$ [duplicate]

Let $A$ be a local ring, $M$ and $N$ finitely generated $A$-modules. Show that if $M \otimes_A N = 0$ then $M=0$ or $N=0$ I read one proof from my book and it goes as follow: First, show that ...
1
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0answers
41 views

Duality for modules

I'm doing exercises given in Maclane "Homology", and I have problems with the following exercise. The task is to state the dual of this proposition: "For submodules $S_t\subset B, \ t\in T$ the ...
1
vote
1answer
40 views

Isomorphism between kernels of R-modules

If $f,g:M\to M$ are homomorphisms of $R$-modules is it true that $\ker(g\circ f)/\ker(f)\simeq \ker(g)$? The morphism $$\ker(g\circ f)\to \ker(g), \ x\mapsto f(x),$$ has kernel equal to ...
1
vote
1answer
37 views

Unusual proof that $M$ is Noetherian if and only if $N$ and $M/N$ are Noetherian

Let $M$ be an $R$-module for a ring $R$. Let $N$ be a submodule of $M$. I read that one can prove that $M$ is Noetherian if and only if $N$ and $M/N$ are Noetherian using these two results: 1) If $M$ ...
2
votes
1answer
58 views

Can a $\Bbb Z$-module be extended to a $\Bbb Q$-module?

Can a $\Bbb Z$-module $M$ be extended to a $\Bbb Q$-module? It is clear that $M$ is now an abelian group. Here $(\frac 1q+ \cdots + \frac 1q)($$q$ times)$m$ = $m$... now from here can I arrive ...
1
vote
1answer
27 views

Universal Linearizer of Alternating Multi-$F[x]$-Linear Maps is Same as that of Multi-$F$-Linear Maps.

Let $V$ be a an $n$-dimensional vector space over a field $F$. Let $M=F[x]\otimes_F V$. We can consider $M$ as an $F[x]$-module by extending scalars using the inclusion $F\to F[x]$. Fact 1. There is ...