Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
2,212
questions with no upvoted or accepted answers
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Non-reflexive module isomorphic to its double dual
Could you give me an example of a non-reflexive module isomorphic to its double dual?
I found an example here but I cannot understand it, do you have any simpler examples?
By this question we should ...
18
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0
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4k
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Every module is the quotient of a free module.
Is every module a quotient of a free module by the following?
Suppose I have a module $M.$ Take the free module $F = \oplus_{m \in M}R.$ Construct a surjective module homomorphism by sending the ...
18
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0
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672
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An elegant description for graded-module morphisms with non-zero zero component
In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the ...
16
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1
answer
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Finitely generated torsion module over a Dedekind domain
Let $M$ be a finitely generated torsion module over a Dedekind domain $R$. Show that there exist nonzero ideals $I_1 \supseteq \cdots \supseteq I_n$ of $R$ such that $M \cong \bigoplus\limits_{i=1}^n ...
11
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0
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146
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Galois connection arising from discussion of flat module and pure exact sequence.
There is somewhat of symmetry in the definition of flat module and pure short exact sequence which can be made precise as follows.
Let $\mathcal{R}$ be the class of all right $R$-modules, $\mathcal{S}$...
11
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153
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How to evaluate Ext of $M/IM$
Let $R$ be a Noetherian commutative ring and $M$ a finitely generated $R$-module. Suppose that $I \subset R$ is an ideal generated by an $R$-regular sequence $\mathbf t = (t_1, \dots, t_r)$ (not ...
11
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0
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570
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A big list - Applications of the structure theorem of finitely generated modules over PIDs
I'm now a TA on an undergraduate course "Algebra II" and the main topics of the course are "rings and modules" and "fields and the Galois theory". We shall cover the ...
11
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0
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565
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Applications of Jordan-Holder theorem in an abelian category
The Jordan-Holder theorem says that any chain of subobjects of a finite lenght object can be refined to a composition series, and that any composition series has the same lenght.
This theorem holds ...
10
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289
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Decomposing the group algebra $\mathbb{R}[Q]$ as a product of matrix rings
I am trying to decompose $\mathbb{R}[Q]$ as a product of matrix rings ($Q$ is the group of quaternions)
By Maschke's theorem, $\mathbb{R}[Q]$ is semisimple. I will begin by decomposing it as a left $\...
10
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788
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Condition for a ring on projective and free modules problem
Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is ...
10
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779
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Proof of Smith Normal Form as a Generalization of Rank-Nullity Theorem
For any matrix $A$ with entries in a PID, there exist invertible matrices $P$ and $Q$ such that $B = PAQ$, where $B$ is in Smith normal form. This theorem is usually proved by using elementary row/...
8
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234
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When does $v_0\wedge\dots\wedge v_{k-1}=0$ when working over a ring that's not a field?
Let $M$ be a module over a commutative ring $R$, and let $v_0,\dots,v_{k-1}$ be elements of $M$. If $R$ is a field then $v_0\wedge\dots\wedge v_{k-1}$ is equal to $0$ if and only if $v_0,\dots,v_{k-1}$...
8
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340
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Geometric intuition for coherent rings, modules, and sheaves
Throughout, all rings are commutative.
Definition 1. A ring $R$ is coherent if the solutions $\mathbf x=(x_1,\dots,x_n)$ to a linear equation $\mathbf{rx}=0$ are a finitely generated $R$-submodule of ...
8
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339
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The importance of being Cohen-Macaulay
I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book.
In that book there are many examples and sentences of the type "If something satisfies this properties, then it is Cohen-...
7
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110
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Is there a ring $R$ for which every ring is isomorphic to the endomorphism ring of some $R$-module?
Is there a ring $R$ for which every ring is isomorphic to the endomorphism ring of some $R$-module (on a fixed side, left or right, so either-or is not allowed)? Here, rings are always required to be ...
7
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923
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Rank of a matrix over a ring?
In his book Module Theory (1977), Blyth defines the column rank of an $m\times n$ matrix $A$ over a commutative unitary ring $R$ to be
the dimension of the subspace of $\mathrm{Mat}_{m\times 1}(R)$ ...
7
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400
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Surjective Endomorphism of Finitely Generated Modules.
Let $A$ be a commutative ring with identity. I would like to prove the following well-known result using a certain approach:
Any surjective endomorphism of a finitely generated $A$-module $M$ is an ...
7
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804
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Jordan Block of Kronecker Product
Let $A$ be a $(p\times p$)-Jordan block of generalized eigenvalue $\lambda$. Let $B$ be a $(q\times q$)-Jordan block of generalized eigenvalue $\mu$. Then what is the Jordan canonical form for $A\...
7
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71
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$M\otimes_A N\cong A$ implies $M$ is left $A$-projective?
Let $A$ be an algebra (possibly non-commutative). Let $M,N$ be $A-A$-bimodules. Suppose that $M\otimes_AN\cong A$. Can we conclude that $M$ is left $A$-projective?
I tried many things but ultimately ...
7
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208
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When is countable direct product of projective modules again projective?
Let $R$ be a commutative ring with unity. The Bass-Papp theorem states that any countable direct sum of injective $R$-modules is injective iff $R$ is Noetherian . Chase's theorem states that any ...
7
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518
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Isomorphisms of free modules and extension of scalars
Let $B$ be a commutative ring with $1$, let $A$ be a subring such that any unit of $B$ which belongs to $A$ is a unit of $A$, and let $\phi:F\to F'$ be a morphism of free $A$-modules such that $B\...
7
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159
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$\mathbb{Z}_m$ as a $\mathbb{Z}_n$-module
It is routine to check that $\mathbb{Z}_m=\mathbb{Z}/m\mathbb{Z}$ is a $\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}$-module if and only if $m \mid n$. This leads to lots of questions such as, "Is $\mathbb{Z}...
7
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184
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Extension of scalars functor essentially surjective.
Let $f:A\rightarrow B$ be a morphism of rings. When is every $B$-module $N$ of the form $M\otimes_A B$ for some $A$-module $M$?. What are sufficient and necessary conditions on $f$? I know for ...
7
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112
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Does $A$ have any special status as an $\mathrm{End}(A)$ module?
Let $A$ be an abelian group and $\mathrm{End}(A)$ its endomorphism ring. Then to give an abelian group $B$ the structure of a (left) $\mathrm{End}(A)$ module, we provide a morphism $\mathrm{End}(A)\to\...
7
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124
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On describing a sort of "well-behaved" subgroups of a free abelian group.
I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case.
Let $M$ be a free abelian group and $N$ a ...
7
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0
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1k
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Computing simplicial homology via Smith Normal Form over Rings
I am not sure whether this is the right forum to ask such a question, if not please let me know.
In the context of my masters thesis, I am working on writing a program to compute simplicial homology ...
7
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0
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1k
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Cardinal of a linearly independent subset of $R$-module
Let $R$ be a commutative ring, and consider $R$ as an $R$-module with the action given by the product of $R$. Prove that if $B\subset R$ is linearly independent, then $\operatorname{card}(B)=1.$
I ...
7
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257
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Symmetric Algebra and Extension of Scalars
I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation
$$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to \mathrm{Mod}(A)$...
6
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0
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76
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Freyd-Mitchell embedding theorem with commutative rings
For a small abelian category ${\cal A}$, the Freyd-Mitchell theorem guarantees that ${\cal A}$ is equivalent to a full subcategory of ${\bf Mod}_R$ for some ring $R$ in a way that preserves exactness. ...
6
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188
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$R$-module with local endomorphism ring is also an $R_P$-module for some $P\in\operatorname{Spec} R$
Let $R$ be a commutative ring, and suppose that $U$ is an $R$-module with local endomorphism ring. (In particular, note that $U$ is indecomposable.) Consider the ring morphism $f:R\to\operatorname{End}...
6
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$R/I\cong R/\text{Ann}_R(R/I)$ but $I\neq\text{Ann}_R(R/I)$
Most of my experience with rings is with commutative rings, so I lack some intuition about the nature of noncommutative rings. Does there exist a noncommutative unital ring $R$ and left ideal $I$ such ...
6
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191
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$R^{\mathbb N}$ as a free $R$-module.
Suppose that $R$ is a commutative ring. I'm wondering if the space $R^{\mathbb N}$ is a free $R$ module.
I know how to prove that it is not a free $R$ module in the case of $R = \mathbb Z$. But the ...
6
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0
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110
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Different "eigenspaces" of a module automorphism with non-trivial intersection
I'm messing around in the following setting: let $$\Bbb C' = \{a+bh \mid a,b \in \Bbb R, h \not\in \Bbb R, h^2=1\} \cong \frac{\Bbb R[x]}{(x^2-1)}$$be the ring of split-complex numbers. Define a ...
6
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0
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207
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When a module over a principal ring is free
Let $R$ be a commutative, principal ideal ring that may have zero-divisors. Is there still a structure theory for the finitely-generated modules over $R$?
In particular, I would like to know a ...
6
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0
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274
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Milnor patching for general modules
The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings
$$
\begin{array}{}
R & \xrightarrow{f_2} & R_2 \\
\downarrow{f_1} & &...
6
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0
answers
610
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Elementary divisors theorem for Dedekind domains (Exercise in Lang's Algebra)
Exercise 13 (b) of Chapter III in Lang's Algebra is as follows.
Let $M$ be a finitely generated projective module over the Dedekind ring $\mathfrak{o}$. Then there exists free modules $F$ and $F^\...
6
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0
answers
1k
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The difference between the ring version and module version of Chinese Remainder Thereom.
Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
6
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0
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273
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Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)
Q: (Exemplification) Do examples family of $R$-modules like $\{M_i\} , \{N_i\}$ and $R$-modules like $M,N$ such that every of four question in underline is true.
(in other word for every question ...
6
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1
answer
221
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A few questions about a specific ring
My question is kinda long, so please bear with me... And I only need hints to get me started.
So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & \...
6
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0
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435
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Category of Chain Complexes of $R$-modules
So I have a couple of questions:
1- Formally speaking, what is a "quotient of a chain complex" of $R$-modules?
2- I want to show that any chain complexes of $R$-modules $C_\bullet $
is the ...
6
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0
answers
125
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Deciding whether or not a class of modules is "big enough"
For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} SA-\...
6
votes
0
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On the Nakayama functor
Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
6
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0
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587
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Characterization of flatness in terms of base change preservation
Let $A \to B$ be a homomorphism of commutative rings and $M,N$ be two $A$-modules. Consider the natural homomorphism of $B$-modules
$\alpha_{M,N} : \mathrm{Hom}_A(M,N) \otimes_A B \to \mathrm{Hom}_B(M ...
5
votes
0
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118
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Subquotients after extension of scalars
Let $M$ be a module over a polynomial ring $k[x_1, \dots, x_n]$ and let $E/k$ be a finite field extension.
In particular, $M \otimes_k E$ is an $E[x_1, \dots, x_n]$-module.
Is it true that every ...
5
votes
0
answers
41
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Counterexample to $\hat{\mathfrak{a}}\hat{M}=\widehat{\mathfrak{a}M}$ when the base ring is not Noetherian or the module $M$ is not finitely-generated
$\def\fra{\mathfrak{a}}$Here it is proven that for $A$ a Noetherian ring, $\fra\subset A$ an ideal and $M$ a finitely-generated $A$-module and if we take $\fra$-adic completions, then $\hat{\mathfrak{...
5
votes
0
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132
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UFD characterization via module theory
While studying module theory, I bumped into some theorems characterizing commutative ring properties through modules :
Let $A$ be a commutative ring with unity, then
$A$ is a field iff every module ...
5
votes
0
answers
117
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Example of an Artinian $B$-Module $M$ and a finite extension $A\subseteq B$ such that $M$ isn't $A$-Artinian
I'm searching an example of a finite ring extension $A\subseteq B$ and an Artinian $B$-module $M$ such that $M$ isn't Artinian as an $A$-module. Note that if such an $M$ is also finitely generated ...
5
votes
0
answers
99
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Determining isomorphism classes of $K[[X]]$-modules
Let $K$ be a field. Determine all isomorphic classes of $K[[X]]$-modules with $K$-dimension 3. For each of these modules $M$, give a $K$-basis and the matrix of the map $M\to M, m\mapsto Xm$ with ...
5
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0
answers
107
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Linear independence of tensors
Let $R$ be a domain with fraction field $K$ and $M$ be a finitely generated $R$-module. If $x_1,x_2,\ldots, x_n\, \in M$ are $R$- linearly independent, then $1\otimes x_1, 1\otimes x_2,\ldots, 1\...
5
votes
1
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124
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example of injective dimension of finitely generated module
Let $R$ be a commutative local ring. $M$ and $N$ are two finitely generated $R$-module of finite injective dimension.
I want to fine an example of $M$ and $N$ such that
$injdim(M)\neq injdim(N)$
Does ...