Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
0
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1answer
30 views
Is a finitely generated module over the field of fractions is also finitely generated over the original integral domain?
Let $R$ be an integral domain and $F$ its field of fractions. Let $M$ be a finitely generated $F$-module.
Question: Is $M$ also a finitely generated $R$-module?
I know that $M$ is an $R$-module ...
0
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0answers
42 views
Question about associated primes and annihilator
I am trying to solve the following exercise:
Let $R$ be noetherian and $M$ a finitely genererated $R$-module. Show that $\mathrm{Ass}(R/\mathrm{Ann}(M)) \subseteq \mathrm{Ass}(M)$ and both sets ...
0
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0answers
14 views
Prove that the following conditions are equivalent for a $_RP$ projective module
Let $_RP$ be a projective module, then:
End($_RP$) is semiperfect
P is semiperfect and finitely generated
are quivalent.
I have to prove this, but I think I'm not understanding the idea behind. ...
0
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1answer
37 views
On covariant linear functors $T: R$-Mod $\to R$-Mod which preserves direct-limits or inverse-limits
Let $R$ be a commutative ring with unity. Let $R$-Mod denote the category of $R$-modules, and $Ab$ denote the category of Abelian groups. Now, it is known that a covariant additive functor
$T: R$-...
1
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1answer
25 views
Does every covariant, additive, faithfully-exact functor $T:R$-Mod $\to Ab$ preserve either direct sum or direct product?
Let $R$ be a commutative Noetherian ring. Let $Ab$ denote the category of abelian groups.
Let $T:R$-Mod $\to Ab$ be a covariant, additive functor such that for any sequence of $R$-modules, $A \...
1
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0answers
24 views
On faithfully flat and faithfully projective modules
Let $R$ be a commutative Noetherian ring. Let $P,Q$ be some $R$-modules such that $-\otimes_R P $ and $ Hom_R(Q,-) $ are faithfully exact functors i.e., for any sequence of modules
$A \xrightarrow{f}...
1
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1answer
10 views
Does taking the radical of modules commute with taking quotients?
I am studying a proof which shows that a particular $R$-module map $\pi$ is surjective onto a module $M$. The details of the map are complex, so I won't give them here, and will just sketch what I ...
1
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0answers
12 views
$(\mathfrak{g},K)$-Module where $K$ is a field?
I am reading through Finite groups of Lie type_ conjugacy classes and complex characters by Roger Carter, and came across this passage where Carter is setting up a special class of module to give a ...
0
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0answers
20 views
The class of left serial rings is closed under extensions
A class $S$ is closed under extension if given an ideal $I \subseteq R$ such that $I\in S$ and $R/I\in S$, then $R\in S$. A ring $R$ is left serial if it is a direct sum of left uniserial rings. ...
1
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1answer
16 views
Is a finitely generated torsion-free R-module free over R if R is an integral domain?
I know this is the case if $R$ is a PID, but PID's are special instances of Integral Domains, so I am wondering if there is a counter-example to the case where R is an integral domain.
This post ...
2
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1answer
38 views
Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective?
Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules ...
1
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0answers
23 views
Local behavior of sheaf of ideals given by a closed immersion
I know that if $Y \hookrightarrow X$ is a closed embedding i of schemes, then the sheaf of ideals
$I_Y(U) = $ {$f \in \mathcal{O}_X(U)\text{ } | i^*(f) = 0$}
is quasi coherent. I sort of understand ...
1
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0answers
36 views
Given $\phi \in \mathrm{End}(M)$, when does $\phi$ injective imply $\phi^*$ surjective and $\phi^*$ injective imply $\phi$ surjective?
Let $M$ be a module over a commutative ring (with unity) $R$. Let $\phi : M \to M$ be an $R$-module homomorphism. Then we have a dual map $\phi^* : M^* \to M^*$ given by $\phi^*(f)=f\circ \phi, \...
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1answer
23 views
Bijective $\mathcal{O}_X$-Module Homomorphisms
Let $(X,\mathcal{O}_X)$ be a ringed space, $\mathcal{F}$ and $\mathcal{G}$$\,\,\,$$\mathcal{O}_X$-modules, and $\varphi:\mathcal{F}\to\mathcal{G}$$\,$ an $\mathcal{O}_X$-module homomorphism.
If $\...
2
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0answers
64 views
Given a split exact sequence $0 \to N \to M \to M \to 0$, when can we say $N=0$?
Let $M$ be a module over a commutative ring $R$. Let $N$ be a submodule of $M$ such that there is a split exact sequence $0 \to N \to M \to M \to 0$. So, in particular, $M \cong M \oplus N$.
Under ...
1
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1answer
13 views
Finitely generated projective resolution of a module over a regular local ring
Let $R$ be a regular local ring and let $M$ be an $R$-module. Then there exists a finite projective resolution $P_\bullet\to M\to 0$. However, need there exist a finite projective resolution ...
1
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1answer
35 views
Hopfian modules and equivalence of categories of modules
For a ring with unity (not necessarily commutative) $R$, let $R$-$Mod$ denote the category of left $R$-modules.
Let $R,S$ be two rings with unity and $T: R$-Mod $\to S$-Mod be an equivalence of ...
2
votes
1answer
30 views
How does Matlis duality behave w.r.t. Hopfian and Co-hopfian modules?
Let $(R,\mathfrak m, k)$ be a Noetherian, complete, local ring. Let $E$ be an injective hull of $k$. We know that the Matlis duality functor $D(-):= Hom_R(-, E)$ gives an anti-equivalence between the ...
1
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0answers
22 views
Torsion-less module over commutative ring whose injective hull is Hopfian
Let $M$ be a module over a commutative ring (with unity) $R$. Let $E_R(M)$ denote the injective Hull of $M$ .
If $M$ is torsion-less (i.e. $\cap_{f\in M^*=Hom_R(M,R)} \ker f=(0)$ ) and $E_R(M)$ is ...
1
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1answer
42 views
Two definitions of torsion pairs/theories. How are they equivalent?
Let $\mathcal{A}$ be an abelian category, $\mathcal{T}, \mathcal{F}$ two strictly full additive subcategories of $\mathcal{A}$. Then according to nLab and other sources including Constructing Torsion ...
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0answers
31 views
A trivialization of line bundle is same as nonvanishing section
I am reading Lemma 17.22.10. My fundamental confusion is how is Nakayama Lemma applied.
Claim: Let $X$ be a ringed space. Assume that each $O_{X,x}$ is a local ring with maximal ideal $m_x$. Let $ ...
1
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0answers
23 views
Endomorphism rings of indecomposable modules
This is a more structured reformulation of this question.
Let $k$ be a field and $A$ be a commutative $k$-algebra, say $A=k[x_1,\dotsc,x_n]$, and $M$ be a finite dimensional (ungraded or $n$-graded) $...
6
votes
1answer
53 views
On Hopfian modules over commutative Noetherian rings
Let $R$ be a commutative Noetherian ring with unity. Let us call an $R$-module $M$ to be Hopfian if every surjective endomorphism $M \to M $ is injective.
1) If $M_1$ and $M_2$ are Hopfian modules, ...
2
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0answers
42 views
Long exact sequence of cohomology
I was reading about cohomology and long exact sequences. I found that
Given $$0 \to L \to M \to N \to 0$$ is a short exact sequence of $G$- modules, then a there exists a long exact sequence is ...
5
votes
1answer
46 views
Is the Artinian property dual to the Noetherian property?
If $R$ is a ring and $M$ is a left $R$-module, we say that $M$ is Noetherian whenever it satisfies any of the equivalent conditions:
1N. Every ascending (under subspace inclusion) chain of submodules ...
0
votes
0answers
32 views
Left Exactness of global sections functor over quasi compact sheaves [duplicate]
Let $0 \rightarrow E' \rightarrow E \rightarrow E''$ be a short exact sequence of quasi coherent sheaves on a scheme X. Show that the sequence $0 \rightarrow E'(X) \rightarrow E(X) \rightarrow E''(X)$ ...
0
votes
1answer
26 views
Proof about length of quotient modules
Let $M$ be an $R$-module and let $x, y \in R$ such that $y$ is not a zero divisor of $M$ and $M / xyM$ has finite length. Show that $l(M/xyM)=l(M/xM)+l(M/yM)$.
In the above, $l$ denotes the lenght ...
2
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1answer
60 views
Is the Evaluation map of an R-Module of rank 1 and hom injective
This is in context of a larger problem of showing that the dual of an invertible sheaf is invertible on a scheme.
I want to show that given a free R-module A of rank 1, the standard evaluation map ...
7
votes
1answer
72 views
Is the ring $R$ a topological ring with respect to the following topology?
Background
This question is motivated by trying to answer this question. But before going into the question straight let me give some background.
Definition 1. Let $R$ be a ring and $A$ be an ...
1
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2answers
50 views
how to prove that $\mathbb Q$ is flat as a $\mathbb Z-$module [duplicate]
I know that $Tor^{\mathbb z}_1(\mathbb Z, N) = 0$ for any $\mathbb Z-$module, because free modules are flat. Then because $Tor_1$ is local, we have $Tor_1^{\mathbb Q}(\mathbb Q, S^{-1}N) = 0$, which ...
1
vote
1answer
34 views
Let $R$ be a local ring with a nilpotent maximal ideal $M$ and $I\subseteq M$. Then $ya\in I$ implies $ya=0$ for some $0\neq y\in R$
Let $R$ be a local ring with a nilpotent maximal ideal $M$. If $I\subseteq M$ is a fixed ideal for which a fixed element $a\in R$, $a+I\neq I$. Prove that for any $0\neq y\in R$ and $y\notin I$, $...
0
votes
1answer
41 views
Indecomposable module with $\operatorname{End}(M)$ non-commutative
Let $A$ be commutative $k$-algebra over a field $k$, and $M$ be a finite dimensional indecomposable* $A$-module. In general, $\operatorname{End}_A(M)/J(\operatorname{End}_A(M)$ will be a division ring ...
0
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1answer
26 views
What is the $\oplus$ sign mean in $R^{\oplus I}$?
I read a defenition of a "finite generated module":
An $R$-module $M$ is said to be finite generated if there exists a surjective homomorphism $R^{\oplus I}\to M$ for some finite set $I$. For $I=\{1,....
0
votes
1answer
36 views
The exact sequence of tensor product
Prove that for all free right $R$-module $F$ and for all exact sequences of $R$-modules $$0\to M\xrightarrow{f}N\xrightarrow{g}P\to 0$$ then
$$0\to F\otimes_RM\xrightarrow{1\otimes f}F\otimes_RN\...
0
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0answers
24 views
A problem about checking isomorphism of R-module
Let $R=K\left[x_1,x_2,\cdots,x_n\right]$ be a polynomial ring with coefficients in the field $K$; $\alpha_1,\alpha_2,\cdots,\alpha_p,\beta_1,\beta_2,\cdots,\beta_q\in R^{1\times m}$; $M_1$ be the $R$-...
0
votes
1answer
25 views
Example of both finitely and infinitely generated free modules which are direct sum of two non free modules
Does there exist non free $R$-modules $F_0,F_1$ such that $F=F_0\oplus F_1$ be a free $R$-module?
1- If yes then for what kind of rings $R$ there exist such $R$ modules?
2- If yes then does it holds ...
1
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0answers
25 views
Determinant of a nonfree module
Is there a definition of a determinant which can be applied to a module with no basis?
We can produce a module with noncommutative rings, without knowing a basis for these rings, i.e. without units. ...
0
votes
1answer
18 views
Condition for a finitely generated flat module be projective
Prove that:
Let $R$ be a commuatative ring, let $T$ be total quotient ring of $R$. A finitely generated flat $R$-module $M$ is projective if and only if the scalar extension $T\otimes_R M$ is a ...
0
votes
1answer
27 views
Why is $M/\operatorname{rad}(M)$ semisimple?
Let $M$ be an $A$-module for $A$ a finite dimensional algebra. Let $\operatorname{rad}(M)=\bigcap\{N\subsetneq M\ \text{maximal}\}$. Clearly, $M/N$ is simple for any maximal submodule $N$. It seems to ...
1
vote
1answer
55 views
Isomorphic algebras of endormporphisms
Let $R$ be a simple algebra, $M$ a simple $R$-module and $N$ a simple $\mathcal{M_n}(R)$-module (always considering finite dimension). Prove that $End_R(M)$ and $End_{\mathcal{M_n}(R)}(N)$ are ...
0
votes
1answer
38 views
Exact sequence construction
Given an $R$-module $M$ arbitrary, show it is always possible to construct an exact sequence of $R$-modules
$$0\longrightarrow K \longrightarrow L \longrightarrow M \longrightarrow 0,$$
with $L$ a ...
0
votes
1answer
23 views
Cardinality of an intersection of two submodules.
Assume $p$ is a prime number and $q = p^2$. Denote by $A$ the ring $\mathbb{Z} / q \mathbb{Z}$.
Consider a finite type module $M$ over $A$ whith cardinality $q^N$ where $N$ is an integer, $N>0$.
...
2
votes
0answers
28 views
Proof that $\Omega_{S^{-1}B/A}=S^{-1}\Omega_{B/A}$
The notations are: $\varphi:A\to B$ is a ring map, $S\subset B$ is a multiplicative subset of $B$, and $\Omega_{B/A}$ is the module of Kähler $A$-differentials of $B$.
In a proof of the fact that $\...
0
votes
1answer
43 views
Invertible $R$-module, $R$ local ring, $L \cong R$
In the last 5 lines of Lemma 17.22.4, the author seems to claim:
If $L$ is an invertible $R$-module, and $R$ is a local ring, then $L \cong R$.
The algebra section doesn't address this case. How ...
1
vote
1answer
26 views
Show that this mapping between localized modules is an isomorphism
Let $R$ be a ring. Let $M$ and $N$ be $R$-modules where $M$ is finitely presented. Then for every multliplicative set $S \subset R$ the canonical mapping
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~Hom_R(M,N) \...
0
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0answers
26 views
Showing that M is a finitely generated R-module
Lemma: Let $A$ be an $R$-Algebra such that $A$ is a finitely generated $R$-Module. Let $M$ be a finitely generated $A$-module. Show that $M$ is a finitely generated $R$-Module.
Proof:
So things I ...
1
vote
1answer
20 views
Ideals of Dedekind rings are projective
Let $R$ be a Dedekind domain and $I$ be an ideal of $R$. Show that $I$ is a projective $R$-module.
My definition of a projective module is that it is a direct summand of a free module, i.e. there ...
0
votes
2answers
50 views
Are the two scalar multiplication on an $R$ module equivalent?
Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. Then, consider $Hom_{\mathbb{Z}}(R, M)$ (note that the homomorphisms are as $\mathbb{Z}$ modules). This is an $R$ module in two ...
2
votes
2answers
45 views
Isomorphic $\mathbb{C}[X]$-modules [duplicate]
My question is :
It is true that $\mathbb{C}[X]/(x-c)$ and $\mathbb{C}[X]/(x-d)$ are isomorphic $\mathbb{C}[X]$-modules if and only if $c=d$?
I have the feeling that the answer is simple but I ...
1
vote
1answer
26 views
If $I\subseteq R$ is a uniserial ideal of $R$ with finite length and $R/I$ is uniserial of finite length, then $R$ is uniserial of finite length
A module is called uniserial if the lattice of its submodules is a chain, i.e., the set of all its submodules is linearly ordered by inclusion. A ring is called right (resp. left) uniserial if it is ...
