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

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38 views

Expressing an $R$-module as a direct sum

Let $R$ be a ring with $1$, $M$ a right $R$-module, and $f:M \to R$ a homomorphism of $R$-modules with $f(M)=R$. Then there is a decomposition $M=K \oplus L$ such that $f \vert_L:L\to R$ is an ...
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0answers
53 views

Finding the structure of an $F_p[X]$module

$p$ is a prime and $M$ is an $F_p[X]-$ module. Given $(X-1)^3M=0, \vert (X-1)^2M\vert=p, \vert (X-1)M\vert=p^3$ and $\vert M\vert =p^7$, determine $M$ as an $F_p[X]-$module, up to isomorphism. Using ...
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2answers
47 views

If $M_1$ and $M_2$ are distinct maximal ideals of, then $R/M_1 \otimes _R R/M_2=0$ [duplicate]

$R$ is a commutative ring with $1$ and $M_1,M_2$ are distinct maximal ideals of $R$. Then $R/M_1 \otimes _R R/M_2=0.$ Consider $f:R/M_1 \times R/M_2\to R/M_1 \otimes _R R/M_2 $ defined by $f(r+M_1,...
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0answers
26 views

Alternative proof : Group algebra contains all irreducible G-modules.

It is well-known that the group algebra $F[G]$ is a direct sum of irreducible $G$-modules. The proof in my text book is as follows Write $F[G] = \oplus_{i=1}^n V_i$, where $V_i$ is a set of ...
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1answer
34 views

Equivalent definitions of semisimplicity of finite dimensional modules

Let $A$ be a finite dimensional $k$-algebra, and let $V$ be a finite dimensional $A$-module. How do we show that $V$ is semisimple (i.e. is the direct sum of simple submodules) if every maximal ...
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1answer
44 views

Modules over $\mathbb Z/p^n\mathbb Z$ have $p$-power order

I am given that $M$ is a module over $\mathbb Z/p^n\mathbb Z$, where $p$ is a prime and $M$ is finite. How can I show that the order of $M$ is $p^j$ for some $j$? The ring $Z/p^nZ$ is not a PID ...
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2answers
80 views

Let $\phi: M \to N$ be an $R$-module homomorphism having a left-inverse $\psi$. Prove that $\ker (\psi) \cong \operatorname{coker} (\phi)$

Let $\phi: M \to N$ be an $R$-module homomorphism having a left-inverse $\psi$. I need to prove that $\ker ( \psi )\cong \operatorname{coker} \phi $. How to approach the problem? Any hints will be ...
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2answers
61 views

Understanding a certain proof about $R$-module homomorphisms and split exact sequences

I'm currently reading "Algebra: Chapter 0" by Paolo Aluffi. Before I state my problem, I would like to give here the definition of split exact sequences from the book: A short exact sequence ...
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0answers
45 views

Minimal spanning set $=$ basis

For finite dimensional vector spaces over a field it is true that if I have a spanning set whose cardinality equals the dimension of the vector space then that spanning set is a basis. Is this theorem ...
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1answer
23 views

Let $M$ be a nonzero cyclic module. $M$ is simple iff $\mathrm{Ann}(M)$ is a maximal left ideal

I'm struggling a bit with the proof of this statement. What I have is that, letting $M$ be a module over the ring $R$ (not necessarily commutative), $\exists \,x\in M$ s.t. $M=Rx$. Now, using the ...
2
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1answer
75 views

$A^\times/k^\times$ is a free $\mathbb{Z}$-module of rank of at most $r - 1$

Consider an algebraically closed field $k$, a finite extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, the integral closure $A'$ of $k[1/T]$ in $K$, and the integral closure $A''$ of ...
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3answers
48 views

Prove that $M/N$ and $N$ are finitely generated $\Rightarrow M$ is finitely generated. [duplicate]

Let $M$ be an $R$-module, and $N$ be it's submodule. I need to prove that if both $M/N$ and $N$ are finitely generated, then so is $M$. How to prove it? Don't have any ideas at the moment.
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1answer
33 views

Proof that primary submodules of $R$ are primary ideals of $R$

I want to prove this: Let $R$ be a commutative ring with identity. If $Q$ is a primary submodule of $R$ (as an $R$-module), then $Q$ is a primary ideal. $Q$ is a primary submodule of $R$ if $r \in R$...
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1answer
45 views

Part of a proof that a left $R$-module $M$ is cyclic and every nonzero element generates $M$ if and only if $M\cong R/I$ for a maximal left ideal $I$.

I'm trying to follow this proof from Noncommutative Algebra by Farb. The theorem is that the following are equivalent for a left $R$-module $M$: $(1)$ $M$ is simple $(2)$ $M$ is cyclic and generated ...
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1answer
48 views

How many ways can $\mathbb{Z}/5\mathbb{Z}$ be given the structure of a module over Gaussian integers?

As the title says, the question is: In how many ways can the additive group $\Bbb{Z}/5\Bbb{Z}$ be given the structure of a module over Gauss integers? My attempt: Any $\Bbb{Z}[i]$-module M is an ...
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1answer
33 views

Prove that an $R$-module $M$ is finitely generated if and only if…

Necesaary definitions: Let $M$ be an $R$-module. We know that $F^R(A) \cong R^{\oplus A}$ where $F^R(A)$ is a free $R$-module on the set $A$ and $R^{\oplus A} = \{ \alpha: A \to R| \alpha(a) \neq 0$...
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2answers
67 views

prove if n - natural number divide number $34x^2-42xy+13y^2$ then n is sum of two square number

prove if n - natural number divide number $34x^2-42xy+13y^2$ where x,y are relatively prime then n is sum of two square number. I don't know what is going on in this exercise. I will be grateful ...
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0answers
35 views

Find smallest vectors in submodule of $\mathbb{Z}^n$

I have a $\mathbb{Z}$ submodule in $\mathbb{Z}^n$ given by a basis. I'm trying to get the smallest elements, where the norm used for smallest is not very important, lets say $\ell^1$ or $\ell^\infty$. ...
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2answers
39 views

Finite generating sets and finitely-generated modules

(All my rings are commutative with $1$.) Reworded a little, a question in a previous Commutative Algebra exam goes like this: Let $A$ denote a ring, $X$ denote an $A$-module $F$ ...
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1answer
32 views

direct sum of noetherian modules is noetherian

We say that an $A$-module $M$ is Noetherian if all of its submodules are finitely generated. Having that definition in mind can anyone give me some hints to prove that if $M$ and $N$ are Noetherian ...
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0answers
34 views

geometric interpretation of eigenfunctions of a vector field

Let $M$ be a smooth manifold and $X\in\mathfrak X(M)$ be a section on the tangent bundle. What is the geometrical interpretation of the eigenfunctions of $X$. That is functions in $f\in \mathcal C^\...
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2answers
36 views

-3 is quadratic residue if and only if $p \equiv 1,7 \pmod {12}$ [closed]

I have to prove that -3 is quadratic residue if and only if $$p \equiv 1,7 \pmod {12}$$ I know one method (with symbol Legendre'a) but I don't get. If someone can explain me I will be happy or give ...
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1answer
34 views

Prove that $3^{(q-1)/2} \equiv -1 \pmod q$ then q is prime number.

$q=2^m+1, m\ge 2$. Prove that if $$3^{(q-1)/2} \equiv -1 \pmod q$$ then q is prime number. I want to use if $q-1 | \phi(q)$, then q is prime number. But I don't know how to transform above equation. ...
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0answers
32 views

Flat module, dependence linear

Anyone help me this question Show that a R-module $M$ is a module flat if only if, such if exist R - dependence linear $\displaystyle \sum a_i m_i =0$ of elements of $M$ , will be exists $a_ij \in ...
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1answer
37 views

Tensor Product and Flat Module

While doing some exercises about flat modules I got to this particular one: Let $I$ be an ideal of $A$ (commutative ring with 1), and $M$ an $A$-module. Show that $I \otimes_{A} M \simeq IM$ if $M$...
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27 views

Prove: If $ p|2^q -1$, p,q primes numbers then p is congruent to 1 modulo q

p|2^q -1, so 2^q - 1 = pk and 2^q = pk - 1 from Fermat we've got 2^q is congruent to 2 (mod q) pk-1 is congruent to 2 (mod q) pk is congruent to 1 (mod q), then k must be 1 Is this evidence ...
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1answer
43 views

extension of derivation of algebras

I am studying extension of derivations, but I am confused of some notations and maybe symbols! For some more details, I recall a theorem. We have the following theorem : Theorem: Let $A$ be an ...
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1answer
42 views

Show $\Bbb{Z}_m\otimes_{\Bbb{Z}}\Bbb{Z}_n=0$ if and only if $m,n$ are coprime. [duplicate]

I'm trying to do this problem out of Atiyah-Macdonald: Show $\Bbb{Z}_m\otimes_{\Bbb{Z}}\Bbb{Z}_n=0$ if and only if $m,n$ are coprime. First, suppose $m,n$ are coprime. Then there exist $s,t$ ...
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1answer
32 views

$M$ is a $\Bbb Z/36 \Bbb Z$-module and $\bar 6 \cdot M = \{0\}$. Prove: $M$ is Noetherian $\iff$ it's Artinian

Suppose that $M$ is a $\Bbb Z/36 \Bbb Z$-module such that $\bar 6 \cdot M = \{0\}$. Prove that $M$ is Noetherian $\iff$ $M$ is Artinian. I managed to prove the forward direction: if $M$ is Noetherian,...
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2answers
35 views

Top exterior product of exact sequence

Let $M,N,P$ be free $R$-modules of rank $a,a+b,b$ respectively, and that they fit into an exact sequence $0\to M\to N\to P \to 0$. Is it true that $\Lambda^{a+b}N=\Lambda^aM \otimes \Lambda^bP$? (...
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1answer
32 views

$\frac{\mathbb{Z}} {p^k\mathbb{Z}}$ as $\mathbb{Z}$-module localized at $(p)$

Consider $\frac{\mathbb{Z}} {p^k\mathbb{Z}}$ as $\mathbb{Z}$-module, where $p\in \mathbb{Z}$ is a prime. What is, up to isomorphism, the localized $(\frac{\mathbb{Z}} {p^k\mathbb{Z}})_{(p)}$ as a $\...
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0answers
30 views

Induced Representations from Tensor Product of Modules

In Chapter 7 of J P Serre's book on Representation Theory, he defines a $\mathbb{C}[G]$-module as $W'= \mathbb{C}[G]\otimes_{\mathbb{C}[H]}W$ and claims it to be the $\mathbb{C}[G]$-module obtained ...
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1answer
12 views

the socle is semisimple

The socle of a module $M$ is semisimple. I know that the socle is $soc(M)=\sum\{N\leq M| \text{$N$ is a simple submodule of $M$}\}$. So $soc(M)=N_1+\cdots+N_k$, where $N_j$ are simple submodules, for ...
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15 views

for a projective module $P$ we have $J(R)P$ is superfluous in $P$ if and only if $\frac{P}{J(R)P}$ has a projective cover.

Let $P$ be a projective module, then $J(R)P$ is superfluous in $P$ if and only if $\frac{P}{J(R)P}$ has a projective cover. Here $J(R)$ is the jacobson radical of $R$ (the underlying ring).
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14 views

Proving $R$ is semisimple if and only if $\frac{R}{J(R)}$ is semiperfect and $R(J)$ and every quotient of $R$ has non-trivial right Socle.

I must prove that a ring $R$ is semiperfect if and only if the following two conditions hold: $\frac{R}{J}$ is semisimple. Every quotient of $R$ has non-trivial Socle. I know that $R$ is ...
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1answer
15 views

Any artinian ring is coherent and perfect.

Let $R$ be an artinian ring. We must prove that $R$ is also a coherent $R$ module. In other words that every finitely generated ideal is finitely presented. To show that $R$ is perfect I think we ...
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0answers
59 views

Can you prove that modules over a field are vector spaces in a categorical way?

Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category (i.e....
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1answer
25 views

Confusion with some theorems leading to the canonical decomposition of an operator

Let $\Bbb K$ be a field and $R = \Bbb K[X]$ be the ring of polynomials with coefficients in $\Bbb K$. Let $\cal L_{\Bbb K}$$(V)$ denote, for a finite dimensional $\Bbb K$-vector space $V$, the set of ...
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3answers
80 views

What does Hom(M,N) mean? Atiyah Macdonald proposition 2.9

In Atiyah Macdonald, "Introduction to commutative Algebra" it says: Proposition 2.9:i) Let $M' \xrightarrow[]{u}M \xrightarrow[]{v} M'' \rightarrow 0$ be a sequence of A-modules and homomorphisms. ...
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1answer
24 views

What is known about finite-dimensional non-semisimple associative algebras over $\Bbb C$?

Artin-Wedderburn says the semisimple ones are sums of matrix algebras, but what about the non-semisimple ones? What are some examples?
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1answer
22 views

Projective Module equivalence proof [closed]

Given an $A$-module $P$. How can I show that if there exists a module $Q$ such that $P \oplus Q$ is free then the functor $\operatorname{Hom}(P,\cdot)$ is exact?
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34 views

Is there a term for the dimension of the annihilator of an element of an algebra?

Let $\mathcal{A}$ be a finite dimensional $R$-algebra, and for $x \in \mathcal{A}$ consider $\mathrm{Ann}(x) = \{ c \in \mathcal{A} \mid cx = 0 \}$. Is there a term for the dimension of this subspace? ...
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1answer
35 views

There always exists a finite, increasing chain of R-submodules of M isomorphic to R/P. Can we describe P?

So I've been studying some commutative algebra and I came across the following theorem Theorem : Let R be a Noetherian ring. Let $M$ be a non trivial $R$-module, finite over $R$. There exists a ...
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0answers
33 views

the field of fractions of a domain $R$ does not have a projective cover if $R$ is not a field.

Let $R$ be an integer domain (so it commutes) that is not a field. Let $K$ be its field of fractions (also called localization). Prove that $K$ does not have a projective cover ( naturally viewed as ...
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0answers
14 views

Appropriate Generalization of Statement about Pure Subgroups to Pure Submodules

I have been working in a book on Homology by Hilton & Stammbach, wherein they introduce the idea of a "pure sequence of Abelian groups", which is a short exact sequence of Abelian groups $$0\...
4
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1answer
45 views

Ring action on another ring?

So a module over a commutative ring $ R$ is an abelian group $G$ equipped with an action given by the product $R\times G\rightarrow G$ that satisfies a few conditions. What if $G$ itself is a ring? Is ...
4
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0answers
89 views

Order topology on $\Bbb Q^+$ as a Z-module

Consider the multiplicative group of strictly positive rationals, denoted $\Bbb Q^+$. This can be viewed as a $\Bbb Z$-module, with the primes serving as a basis. If we place the order topology on $\...
1
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1answer
102 views

Question concerning modules over a Clifford algebra

Let $R$be a commutative ring with unit element, and $M$ be an $R-$module. Let $f:M \times M \to R$ be a nondegenerate symmetric bilinear quadratic form, and $C(f)$ be the corresponding Clifford ...
1
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0answers
16 views

definition of Kasparov modules, what is a degree 1 operator?

I read the book "K-theory for operator algebras" and I have a question about the definition of Kasparov modules for graded $C^*$-algebras $A$ and $B$. Definition: Let $A$ and $B$ graded $C^*$-...
1
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1answer
67 views

How can we find the elementary divisors?

We consider the polynomial ring $R=\mathbb{R}[t]$ and the $R$-module $M=\mathbb{R}^3$, where $a\cdot x$ ($a\in R,x\in M$) is defined as usual if $a\in \mathbb{R}$, and $a\cdot x=(x_1, 0, 0)$ if $a=t, ...