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

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1answer
40 views

Proof of the exactness of the tensor product

In Atiyah and MacDonald, Prop 2.18 establishes that for any exact sequence $$M'\xrightarrow{f}M\xrightarrow{g}M''\xrightarrow{}0\tag{1}$$ of $A$-modules and homomorphisms, and for any $A$-module $N$, $...
2
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2answers
63 views

Proving/Disproving $M$ has the structure of an $R$-module

Given an abelian group $M$ and a ring $R$, how can one prove or disprove that $M$ has the structure of an $R$-module? When proving $M$ is an $R$-module, if it is not obvious how to define an action $R\...
1
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1answer
48 views

On boolean algebras as rings, modules and/or R-algebras

After trying to make sense of first order logic from an algebraic point of view I started to read about boolean algebras (similar to the explanations given here: wikipedia on boolean algebras. I also ...
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2answers
22 views

Element separated from a submodule by a homomorphism

Let $M$ be a module over a commutative ring $A$, $M'$ a submodule, and $y\in M\setminus M'$. Then $y$ can be separated from $M'$ by homomorphism. What I wish to prove is that there exists an $A$-...
2
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1answer
38 views

Socles and factors

Let $A$ be a finite dimensional algebra over a field $K$ and let $M$, $M'$ and $N$ be $A$-modules. Suppose that $M'\subseteq M$ and assume that $N$ has simple socle. Let $f: M \longrightarrow N$ be ...
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0answers
28 views

About rank of free module compared to spanning and LI set

Let M be a free R-module where R is commutative ring with unity. Q1: If M is FG by n vectors. Will rankM $\leq$ n? Q2: If M has a LI set $S$. Will #$S \leq$ rankM? What if R is PID? Are above ...
1
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1answer
27 views

Writing a homomorphism in multiplicative form

Consider the additive group $\mathbb {Z_2}=\{0,1\}$ and consider the group $\mathbb Z_2^n$. Let $u_1,\ldots,u_d$, $(d>n)$ be not necessarily distinct , non-zero elements of $\mathbb Z_2^n$. Define ...
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0answers
18 views

Can we know what are the form of submodules of Z\oplus Z. [duplicate]

I know for instance Z(2,3), Z(2n,0) or Z(n,n) are submodules of the Z-module Z+Z. But can we know what are the submodules of Z\oplus Z exactly?
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1answer
15 views

Relation between the dual module and the dual of a vector space.

I need to know if there is a relation between the dual module of a subspace U of a finite dimensional vector space V, looked as a G-module, and the dual vector space of U. In this case, G is a finite ...
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4answers
54 views

Why is a bilinear map $M \times N \to B$ the same as a homomorphism $M \to Hom_R(N,P)$?

Eisenbud's "Commutative Algebra" states that: It is elementary that a bilinear map $M \times N \to P$ is the same as a homomorphism $M \to Hom_R(N,P)$ While this makes some intuitive sense I keep ...
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0answers
27 views

A embedding of tensor product over semisimple algebras

Let $R$ be a semisimple Artinian algebra over the complex number field $\mathbb{C}$, that is, $R$ is isomorphic a finite direct product of matrix rings over $\mathbb{C}$. Let $S$ be a ideal of $R$, ...
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1answer
57 views

Compute the projective dimension of the given $R$-module

Let $$R=\frac{K[[x,y,z]]}{\left<xz,yz\right>}\text{ and } M=\frac{R}{\left<z+\left<xz,yz\right>\right>}.$$ Compute the projective dimension of $M$ as an $R$-module. My attempt ...
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2answers
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 ...
2
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0answers
55 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 ...
1
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2answers
52 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 ...
1
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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 ...
0
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1answer
45 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 ...
2
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2answers
85 views

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

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 ...
1
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2answers
63 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 ...
1
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1answer
24 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
votes
1answer
87 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
49 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.
0
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1answer
35 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
47 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 ...
1
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1answer
49 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 ...
1
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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 ...
2
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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 $F$ ...
2
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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
38 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
37 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 ...
2
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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
33 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 ...
1
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1answer
41 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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0answers
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 ...
0
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1answer
44 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 ...
1
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1answer
47 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
36 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 $\Lambda^{a+b}N=\Lambda^aM \otimes \Lambda^bP$? (...
1
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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
33 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 ...
1
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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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0answers
15 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 ...
2
votes
1answer
16 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 ...
2
votes
0answers
60 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....
0
votes
1answer
27 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
83 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. ...