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

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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 ...
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136 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
24 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
8 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 ...
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1answer
27 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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14 views

Induced homomorphism between quotient modules

Let $R$ be a commutative ring with identity. Let $M,N$ be $R$-modules, and let $I$ be an ideal in $R$. Let $\varphi : M \to N$ be a homomorphism such that the induced homomorphism $$\bar{\varphi} : ...
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29 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 ...
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1answer
23 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
12 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
70 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 ...
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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
49 views

motivation for the direct limit [on hold]

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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13 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
14 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 ...
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1answer
22 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 $i \neq j$ ...
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17 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) = ...
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1answer
21 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 ...
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1answer
19 views

Poincare series and Hilbert polynomial of some graded modules

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, ...
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1answer
49 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 ...
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1answer
24 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 ...
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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 ...
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21 views

Reference request for numerical invariants of modules which are not finitely generated

Suppose that $R$ is an integral domain with subring $S$ and that both rings are finitely generated $k$-algebras ($k$ an algebraically closed field). $R$ is integral over $S$ if and only if $R$ is ...
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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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25 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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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 ...
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31 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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52 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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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
41 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 ...
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1answer
48 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 ...
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1answer
54 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 ...
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1answer
20 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 ...
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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
37 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 ...
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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 ...
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1answer
29 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 ...
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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 ...
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1answer
37 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 ...
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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 $M$ ...
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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 ...
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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 ...
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1answer
22 views

Invariant complement to a $G$-module (not necessarily a vector space)

Let $G$ be a group, $R$ a ring (not necessarily a field), and $M$ an $R$-module. Assume we have a group action $\rho:G \times M \to M$. If there exists a $G$-invariant submodule $N \subseteq M$, is ...
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1answer
36 views

How to think of $R[x,y]$-modules?

There is a very concrete description for $R[x]$-modules: they correspond "nicely" with $R$-modules equipped with an endomorphism, by the following: On the one hand, an $R[x]$-module $M$ is in ...
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0answers
49 views

Subtle error with a module endormorphism on $\mathbb{Z}_8 \times \mathbb{Z}_8$

Let $a,b,c$ be arbitrary integers such that $a$ is odd and $(a,b,c)=1$. Let $R = \mathbb{Z}_8$, the set of all integer residues modulo $8$. Define an $R$-module endomorphism $\phi \colon R \times R$ ...
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21 views

Can one add reasonable assumptions that we have $depth\ R \geq depth\ M$ for every $R$-module $M$?

Let $(R,m)$ be a commutative Noetherian local ring which is not CM. Let $M$ be a finite $R$-module. Here, Hanno shows that one can have any inequality between $depth\ R$ and $depth\ M.$ Still, the ...
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1answer
21 views

Example of module homomorphism from torsion to free

Can someone give an example of an R module homorphism from a torsion mkdule N to free module M where R is commutative with unity with the homomorphism different from the zero one I know if we dont ...
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2answers
76 views

Determinant from Paul Garret's Definition of the Characteristic Polynomial.

$\DeclareMathOperator{\id}{id} \DeclareMathOperator{\End}{End}$ On pg. 390 of Paul Garret's notes on Algebra, a definition for the characteristic polynomial is given, which I discuss here. Let $V$ be ...
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0answers
8 views

Relationship between nonnegative real semiring module and boolean semiring module

In nonnegative matrix factorization, one attempts to factor a matrix $\mathbf{X} \in \mathbb{R}_{\geq 0}^{m \times n}$ into matrices $\mathbf{Z} \in \mathbb{R}_{\geq 0}^{m \times k}$ and $\mathbf{A} ...
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1answer
34 views

Find the structure of $\mathbb{Z}_{120}^*$

How to find the structure (in term of cyclic groups) of $\mathbb{Z}_{120}^*$? I know that the number of elements of $\mathbb{Z}_{120}^*= \phi(120) = 32 = 2^5$ But then, any hints?