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

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5
votes
3answers
119 views

Properties characterized by a vanishing Ext or Tor module

While reading Weibel's "An introduction to homological algebra'', I've noticed that many properties of a module are characterized by the vanishing of some Tor or Ext. Fix a (commutative) ring $R$ and ...
4
votes
2answers
37 views

Easy proof from Atiyah-McDonald on module homomorphisms

Let $v \in \text{Hom}(M,M')$ Then if the induced hom $\bar{v} : \text{Hom}(M',N) \to \text{Hom}(M,N)$ given by $f \mapsto f \circ v$ is injective for all $N$, then $v$ is surjective. Attempted proof ...
2
votes
0answers
22 views

a question about division ring and injective modules

the theorem 4.10 iii) in the page 204 of Hungerford say: if $R$ is a división ring and $0\rightarrow A\stackrel{\theta}{\rightarrow}B\stackrel{\zeta}{\rightarrow}C\rightarrow 0$ is a short exact ...
1
vote
1answer
22 views

Why there is a less care of symmetricity of bimodules over a commutative ring?

Let $R$ be a commutative ring. Then, we say $M$ is an $R$-module instead of left $R$-module or right $R$-module or $(R,R)$-bimodule. I'm curious why this convention is acceptable in general. Let's ...
0
votes
1answer
45 views

Prove or Disprove: Finitely generated Artinian module is Noetherian.

I think it is true and I am trying to prove it. I am considering reducing the case to Artinian rings. Say $M$ is finitely generated Artinian $R$-module. Then $R/Ann(M)$ is an Artinian $R$-module. Thus ...
2
votes
1answer
37 views

Is every integral ring extension finite?

I think it is not necessarily the case. So far I have that if $S\subset R$ is a an integral infinite ring extension, we would need to have that $R$ is infinitely generated but $S[R]$ is finitely ...
0
votes
0answers
17 views

Decomposition of kernel of an epimorphism

Let $f:P→M$ be a projective cover of a left $R$-module $M$ in the sense that $f$ is an $R$-epimorphism from a projective $R$-module $P$ to $M$ with a small kernel in $P$. Assume $P=P_1⊕P_2$ so that ...
1
vote
1answer
36 views

Nontrivial map between finite modules over local noetherian ring

I have just read that over a noetherian local ring $(A,\mathfrak{m})$, there is always a nontrivial map between two finitely generated modules such that their support is exatly $\mathfrak{m}$. The ...
1
vote
1answer
26 views

R-module structure from R-algebra

I am reading through Dummit and Foote and one of the examples I do not understand. The example is from 10.1, page 343. (3) If $A$ is an $R$-algebra then the $R$-module structure of $A$ depends only ...
1
vote
1answer
61 views

About submitting a paper in algebra?

My questions are about submitting papers related to modules category: (1) how much does it take (usually) to get the acceptance from a journal with good impact factor? (2) which journals(with good ...
0
votes
1answer
9 views

How do I show the basis of the tensor product is of this form?

Let $R$ be a non-trivial commutative ring, hence $R$ has IBN property. Let $M$,$N$ be free $R$-modules. Then the tensor product $M\otimes_R N$ is free and $rnk(M\otimes_R N)=rnk(M)rnk(N)$. Let ...
1
vote
1answer
38 views

How to classify all the abelian groups with finite exponent?

Let $A$ be an abelian group, the exponent exp$A$ is the least natural number $n$ (if exists) such that $nA=0$ or $+\infty$. The question can be reduced to the case exp$A=p^n$ for a certain prime ...
1
vote
1answer
25 views

Does this theorem assume the commutativity?

I'm reading this article : https://drexel28.wordpress.com/2012/01/25/tensor-product-of-free-modules/ I don't get a theorem in the above page: Theorem: Let $M$&$N$ be free left $R$-modules. ...
2
votes
1answer
55 views

Module in which every submodule containing the radical is an intersection of maximal submodules.

$\newcommand{\Rad}{\operatorname{Rad}}$Let $M$ be a left $R$-module with $\Rad(M)\neq 0$ where $\Rad(M)$ is defined as the intersection of all maximal submodules of $M$. Thus, if $K = \bigcap_{i\in I} ...
3
votes
1answer
31 views

Does free bimodule exist?

Let $R,S$ be rings and $A$ be a set. Does there exist a free $(R,S)$-bimodule $F(A)$ on $A$? How do I construct it? Is it just $\oplus_{i\in A} (R\times S)$?
9
votes
0answers
43 views

Why are noetherian and artinian modules important?

As a TA I was recently asked to give the students an introduction to two (quite related) concepts that are new to me, noetherian and artinian modules. I intend to prove the characterisation theorem ...
-1
votes
1answer
46 views

Direct sum over ring [closed]

Let $R$ be a PID and $F$ be a free module with basis $\{e_1, e_2\}$. Let $u = me_1 + ne_2$ for $m,n \in R$. Prove that $Ru$ is a direct summand of $F$ if and only if the ideal $(m,n)=R$.
0
votes
0answers
19 views

Any cyclic module can be given an algebra structure?

Given a cyclic module $M$ over a commutative ring $R$, we know that $M$ is isomorphic (not canonically) as an $R$ module to $R/A$, where $A$ is the annihilator of $M$ in $R$. But $R/A$ is an $R$ ...
0
votes
0answers
25 views

How do I generalize this theorem?

Dummit&Foote p.362 Let $R$ be a subring of $S$ and let $N$ be a left $R$-module and let $\iota:N\rightarrow S\otimes_R N:n\mapsto 1\otimes n$ be an $R$-module homomorphism. Let $L$ be ...
0
votes
0answers
18 views

What's the point of the extension of scalars?

Let $R,S$ be rings and $f:R\rightarrow S$ be a ring homomorphism. Then we give $S$ a right $R$-module structure as $s•r =sf(r)$. It seems natural to give an $R$-module structure in this way, but I ...
1
vote
2answers
84 views

An abelian group A is torsion iff A ⊗ Q = 0

At the moment I know the fact for finitely generated abelian groups and this: If $x \otimes y = 0 \in M \otimes N$, then there is a finitely generated submodule $L \subset M$ containing $x$ such that ...
2
votes
2answers
46 views

Structure theorems for modules over 'good' rings

Structure theorem for finitely-generated modules over PID is well-known fact. But is there similar theorems for modules(maybe finitely-generated) over noetherian or artin or some other 'good' rings? I ...
1
vote
1answer
24 views

Direct sum of free module?

Let $R$ be a PID, and let $F$ be a free module with basis $e_1, e_2$. Let $u\in F$ such that $u = me_1 + me_2$ where $m \in R$. How can I show that $Ru$ is a direct summand of $F$ if and only if $m$ ...
0
votes
2answers
32 views

Example to show that multiplication by ideals and intersection of submodules do not commute

The key point of question about typical proof of Krull Intersection Theorem is that multiplication by ideals and intersection of submodules do not commute. Can anyone give me an example of this? ...
0
votes
0answers
31 views

Does the abelian group in the definition of a left R-module need to be additive?

Let $R$ be an arbitrary ring and $M$ an abelian group, a left $R$ module consists of the aforementioned abelian group $M$ and an operation $f:R \times M\rightarrow{M}$ satisfying for $r,s\epsilon{R}$ ...
4
votes
2answers
55 views

Dimension of irreducible module divides the dimension of the algebra?

Fact: $\chi(1)$ divides order of $|G|$ where $\chi$ is an irreducible character of $G$. Above fact is equivalent to say that if $V$ is an irreducible $A=\mathbb C [G]$ module then $\dim(V)$ divides ...
0
votes
1answer
22 views

Example of module homomorphism?

I would like to come up with an example for a module homomorphism between two finitely generated, not free modules over $\mathbb{Z}[i]$. Also, what would be the kernel and image in this example?
1
vote
1answer
62 views

Intersection of submodules

I have question regarding intersection of submodules. Could anyone give example of a commutative ring $R$ with identity and an $R$-module $M$ such that $$IM\cap JM\nsubseteq (I\cap J)M$$ for some ...
2
votes
0answers
43 views

$\mathbb{Z}_p$ as a module over $\mathbb{Z}_{(p)}$

I denote by $\mathbb{Z}_{(p)}$ the localization at a prime p, and by $\mathbb{Z}_p$ the p-adic integers. Question: what is the structure of $\mathbb{Z}_p$ as $\mathbb{Z}_{(p)}$- module? For example ...
1
vote
1answer
28 views

Direct sum is isomorphic?

Given R, a PID, and two finitely generated R-modules A and B, suppose $\varphi: A \rightarrow B$ is a homomorphism. If B is a free module, show that $A \cong \ker(\varphi) \oplus ...
3
votes
1answer
63 views

Show an ideal is a finitely generated projective module via a split exact sequence

Let $I$ be an ideal of $R$ such that the mapping $f:I\otimes_R\operatorname{Hom}_R (I,R)→R$ defined (on the generators) by $f(i\otimes α)=α(i)$ for all $i∈I$ and $α∈\operatorname{Hom}_R (I,R)$ is ...
0
votes
0answers
51 views

A property for finitely generated modules

Consider the folowing property for an $R$-module $M$ (where $R$ is an associative ring with identity): For each family $\{M_{\alpha}\}_{\alpha\in I}$ of $R$-modules and each homomorphism $f:M\to ...
0
votes
0answers
19 views

Closure properties for classes of modules that form a cotorsion pair

A torsion theory is a pair of classes of $R$-modules (where $R$ is an associative ring with identity) $({\mathbb T},{\mathbb F})$, such that $r({\mathbb T})={\mathbb F}$ and $l({\mathbb F})={\mathbb ...
2
votes
3answers
95 views

Does a long exact sequence of flat modules remain exact after tensoring with an arbitrary module?

In Liu's Algebraic Geometry and Arithmetic Curves, Proposition 1.2.6 states that given any short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ with $M''$ flat, taking ...
1
vote
1answer
18 views

Superfluous condition on universal property of tensor product

Let $R$ be a ring and $M$ be a right $R$-module and $N$ be a left $R$-module. Let $\phi:M\times N \rightarrow M\otimes_R N$ be the canonical middle linear map. Here is a theorem in my text under the ...
0
votes
0answers
12 views

Characterisation of module structure on the tensor product

Let $S,R$ be rings and $M$ be an $(S,R)$-bimodule and $N$ be a left $R$-module. Then, there exists an openration $•:S\times (M\otimes_R N)\rightarrow M\otimes_R N$ such that $s•(m\otimes n)=sm\otimes ...
1
vote
1answer
38 views

An inverse limit exact sequence for complete modules

Let $A$ be a commutative complete ring with unit for the $I$-adic topology, where $I$ is the ideal of $A$. Let $(M_n)_{n\geq 0}$ be $A$-modules such that $I^{n+1}M_n=0$ and that there exist a ...
2
votes
1answer
51 views

Extending scalars from $k$ to $K$: how find $K$ linear maps?

If I extend scalars from a ring $R$ to a ring $S$ by a homomorphism $f:R \to S$, then starting with an $R$ module $M$, I get an $S$ module $S \otimes_R M$. Given $\sigma \in \text{End}_R(M)$, I know ...
3
votes
1answer
53 views

Quotient of modules

I' m having trouble with the following exercise: Let $Y$ be a submodule of $X$ and $X$ a submodule of $Z$ (modules over a ring $R$). Suppose that $X/Y$ is isomorphic to $Z$. I need to conclude ...
2
votes
1answer
61 views

Equivalence of categories involving graded modules and sheaves.

Let $S$ be a graded ring with $S_0=A$ a finitely generated $\mathbb{K}$-algebra and $S_1$ a finitely generated $A$-module. Let $M$ be a graded $S$-module and $\tilde{M}$ the corresponding sheaf on ...
0
votes
2answers
41 views

finite dimensional $K$-vector space $V$ with linear endomorphism $T$ is a cyclic module over $K[x]$

Problem: Suppose that $V$ is a finite dimensinal $K$-vector space with a linear endomorphism $T$. Then show that i) Associated $K[x]$-module $V$ is such that $V\cong K[x]/(g)$ for some monic ...
1
vote
2answers
42 views

Does tensoring the integers $\Bbb Z$ with any field $F$ give that field itself? How to prove that?

Does tensoring the integers $\Bbb Z$ with any field $F$ give that field itself? How to prove that? I know for $\Bbb Q$ it is true.
1
vote
1answer
12 views

Definition of the tensor product of finite sequence of modules

I have posted several questions about the tensor product of modules before and this post would be the final one. I have read wikipedia,mathSE,Dummit&Foote and Bourbaki for the definition of the ...
0
votes
0answers
17 views

Definition of multimodule

Reference : Bourbaki - Algebra I p.224 Let $\{A_i\}_{i\in I}$ and $\{B_j\}_{j\in J}$ be families of rings. Let $M$ be a set such that for each $i\in I, M$ is a left $A_i$-module and for each ...
0
votes
1answer
62 views

If $V$ is a vector space to itself, is the idempotent linear transformation, prove there is a basis such that is a diagonal matrix with all $0$ or $1$

Im trying to solve an algebra homework problem, basically i have: $$\phi:V \longrightarrow V$$ such that $\phi = \phi^2$, where $V$ is finite dimensional. I already proved that $V = \phi(V) \oplus ...
0
votes
1answer
43 views

Faithfully Flat Abelian Groups

I need some help to find faithfully flat abelian groups. Flat abelian groups are torsion free $\mathbb{Z}$-modules. But what about faithfully flat abelian groups. $\mathbb{Q}$ is an example that is ...
1
vote
1answer
31 views

Definition of tensor product

Here is the standard dedonition of the tensor product of two modules: Definition for tensor products of two modules: Let $R$ be a ring and $M$ be a right module and $N$ be a left module. ...
1
vote
0answers
28 views

Simplified Exponent in $\bmod$ equation

I am trying to simplified the following expression: $$\left(y^m\right)^{x} \bmod p$$ In my case, I can only solve $(y^x) \bmod p$ first without prior knowledge of $m$. Eventually, my answer should ...
0
votes
0answers
36 views

What is $\sum dim(H_i(X,F))$?

Let $X$ be a finite CW complex and $F$ be any field. Then what is dimension of $\sum_iH_i(X,F)$? This question has three sub parts as far what I think i) if $F$ is of char $0$ then $\Bbb Q \subseteq ...
1
vote
0answers
28 views

Is there less use of infinite rank modules than finite rank modules?

Let $R$ be a nonzero ring and $M$ be a free module on $R$ and here are theorems I know. If $M$ has a infinite basis, then rank of $M$ is well-defined If $R$ has IBN property, then rank of $M$ ...