Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
5,617 questions
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Calculating the dimension of the tensor product of modules
This is a recent qual problem that I am struggling with.
Put $M=\mathbb{C}[x]/(x^2+x)$ and $N=\mathbb{C}[x]/(x-1)$.
$(a)$ What is the dimension of $M \otimes_{\mathbb{C}[x]}N$ as a vector ...
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2answers
22 views
Is a submodule of a free module also free? [duplicate]
Is a submodule of a free module also free ? For me it looks natural that yes, but in my course it's written that it's only true for a module over a PID and I don't really understand why. Any example ?
5
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1answer
68 views
$H \leq \mathbb{Z}_q^n$ and $H \cong \mathbb{Z}_q^m$ implies that $\mathbb{Z}_q^n / H \cong \mathbb{Z}_q^{n-m}$
Given a pair of positive integers $n,q$ and a subgroup $H \leq \mathbb{Z}_q^n$ such that $H \cong \mathbb{Z}_q^m$ for a positive
integer $m < n$ then show that
$$ \mathbb{Z}_q^n / H \cong \...
1
vote
1answer
14 views
On the two definitions of Rings of Quotients in T.Y.Lam
In the book "Lectures on Modules and Rings" by T.Y.Lam there are two definitions:
((8.2) Definition, T.Y.Lam.) We say that $N$ is a dense submodule of $M$ (written as $N\subseteq_d M$) if, for any $...
1
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1answer
59 views
+200
Short proof of $\mathbb{Q}[x,y]/\langle x^2+1, y^4-2\rangle \equiv\mathbb{Q}[\sqrt[4]{2}, i]$
I am looking for an indirect proof of $$E = \mathbb{Q}[x,y]/_{\langle x^2+1, y^4-2\rangle}\cong\mathbb{Q}[\sqrt[4]{2}, i],$$ much preferably using module homomorphism theorems.
To be more specific, ...
3
votes
1answer
48 views
Proving that $\operatorname{Tor}_n^R$ is a bifunctor
$\newcommand{\Tor}{\operatorname{Tor}}$
Ex10.2 pg 615: For a ring $R$ and fixed $k \ge 0$, prove that $\Tor_n^R(-,-)$ is a bifunctor.
I am aware of this post. I am also not satisfied with the ...
2
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0answers
34 views
surjective morphism between $\mathbb{Z}$-modules
Let $q$ be an integer, then there is a one to one corrispondence between $\hom_{\mathbb{Z}_q}(\mathbb{Z}_q^m, \mathbb{Z}_q^n)$ (in my case $m > n$) and the matrices $\mathbb{Z}_q^{n \times m}$.
A ...
0
votes
0answers
18 views
Is a ring R a progenerator for the category R-mod?
I have to demonstrate that $R$ viewed as left $R$-module is a progenerator (i.e. a projective and finitely generated generator) of $R$-mod, the category of left $R$-modules.
How can I do this?
I know ...
2
votes
2answers
27 views
Is $(x)$ as a $k[x]$ module free?
Is $(x)$ as a $k[x]$ module free?
I think it is free because it seems the basis element is $x$ and it is not annihilated by any element of $k[x]$.
Thanks!
1
vote
1answer
32 views
Cardinality of generated rings and generated modules
I've once asked a similar question only about groups, but I am interested whether the logic is still sound:
$(1)$Let $S$ be a generating set of a ring $R$, and denote $\kappa=\vert S\vert$. Then $\...
4
votes
1answer
31 views
Question involving characteristic polynomial of a linear transformation
I was wanting some hints on a question and I have no idea how to approach this:
Suppose $F$ is a field, $V$ is an $F$-vector space and $T: V \rightarrow V$ is a linear map. Suppose $p(x) \in F[x]$ ...
0
votes
0answers
14 views
maximal $R$-linearly independent subset of a finitely generated module
Let $M$ be a finitely generated $R$-module with rank $n$, and let $N = \left\{ y_{1},\ldots,y_{m}\right\}$ be an $R$-linearly independent subset of $M$, where $m \leq n$.
My question is simple: Is it ...
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0answers
37 views
Prove that the image of a ring via functor is a bimodule
I'm studying Morita theory on "Rings and Categories of Modules", Anderson.
I have some problems with a Theorem about equivalent rings.
Let $R$ and $S$ be equivalent rings via inverse equivalences $F: ...
1
vote
1answer
44 views
A left $A$-approximation of a torsionless module
Let $A$ be an Artin algebra. A finite dimensional $A$-module $M$ is torsionless, that is $M$ can be embedded into a projective $A$-module. We say a map $f: M \rightarrow P$ a left $A$-approximation of ...
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1answer
19 views
$rad(P)$ non-projective induces all submodules of P non-projective?
Let $A$ be a finite dimensional algebra. $P$ is an idecomposable projective $A$-module such that its radical $rad(P)$ is non-projective.
Is it right that every non-zero proper submodule of $P$ is not ...
1
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1answer
50 views
Does the definition of “local module” need to say all proper submodules are contained in the maximal submodule?
I would like to know: Is there any definition for local modules like definition for local rings?. In other words: what is the name of a module with only one nontrivial maximal submodule?
1
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1answer
25 views
Splitting of Algebra Acting on a Module?
Let $A$ be a finite dimensional commutative unital algebra over $\mathbb{K}$ (of most interest to me in $\mathbb{K}=\mathbb{R}$). Let $V$ be a finitely generated $A$-module and let $m:A\otimes_{\...
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0answers
36 views
Is every submodule contained in a maximal submodule (assuming there are two max. sub.)?
Let a module $M$ has exactly 2 maximal submodules say $M_1$ and $M_2$. Then is it true to say that ''every proper submodule of $M$ is contained in $M_1$ or $M_2$?''
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1answer
45 views
Is it true to say that : Every submodule of a module M contains in a maximal submodule?
Let $R$ be an arbitrary ring with $1\neq 0$ and $_RM$ a left $R$-module. Is it true to say that : Every proper submodule of a module $M$ is contained in a maximal submodule?
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0answers
32 views
Exercise on block theory of finite dimensional algebras
I am finding some problems in solving this exercise.
Assume $A$ is a finite dimensional $K$-algebra, with $K$ an algebraically closed field. Let $B_1,\dots, B_r$ be the blocks of $A$, with unit ...
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37 views
check that a module over polynomial ring is free
I encountered this problem when I was doing practice exercises on Modules over PID:
Let $F$ be a field with characteristic $\neq 2$,
and let $V$ be a vector space over $F$ with basis $\left\{v_{1},...
5
votes
1answer
73 views
Why does Herstein add all these extra hypotheses for a “simple” modules proof?
An exercise paraphrased from Herstein's Topics in Algebra (2nd edition, Chapter 4, §4.5, problem 12):
Let $M$ be an irreducible left $R$-module, where $R$ is an arbitrary ring and $rm \neq 0$ for ...
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0answers
25 views
Definition of socle of a module
For me, the socle of an $R$-module $M$ is the unique maximal semisimple submodule of $M$. If $(R,\mathfrak m)$ is a local ring, this is equivalent to the maximal submodule annihilated by $\mathfrak m$....
2
votes
1answer
35 views
Isomorphism between $R$-algebra $RG$ and $RG^{\ast}$
Let $R$ be a commutative ring, $G$ be a finite abelian group. Consider a group ring $RG$ as an $R$-coalgebra. Is it true that $RG\simeq RG^{\ast}$ as an $R$-algebra? If the answer is true, please tell ...
3
votes
1answer
47 views
If $R$ is a ring and $R^n$ and $R^m$ are isomorphic as left $R$-modules then they are also isomorphic as right R$-modules
Is it necessarily true that if $R$ is a ring and $R^n$ and $R^m$ are isomorphic as left $R$-modules then they are also isomorphic as right $R$-modules.
It appears as if they are.
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0answers
25 views
Finitely generated module $M$ over principal ideal domain splits into direct sum of $M_\text{tors}$ and $M' \subset M$.
Let $R$ be a principal ideal domain and $M$ be an finitely generated $R$-module.
The torsion module $M_{\text{tors}}$ is defined as
$$
M_{\text{tors}} := \left\{ m \in M \;|\; \text{there is}\; a\in ...
3
votes
1answer
74 views
How to count the number of elements in $(\mathbb{Z}[i]/I^{2014})\otimes_{\mathbb{Z}[i]}(\mathbb{Z}[i]/J^{2014})$?
Let $I,J\unlhd \mathbb{Z}[i]$ be the principal ideals generated by $7-i$ and $6i-7$, respectively. Find the number of elements in the $\mathbb{Z}[i]$-module $A=(\mathbb{Z}[i]/I^{2014})\otimes_{\mathbb{...
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0answers
44 views
Abelian $\operatorname{Hom}(-,X)$ in $\mathcal{Alg}_{R/A}$
How do I find all objects $X$ such that $\operatorname{Hom}(-,X)$ is an abelian group in the category $\mathcal{Alg}_{R/A}$ of algebras over a fixed $R-$algebra $A$?
I have been considering what ...
3
votes
0answers
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Differentials on the second page of the spectral sequence of a first quadrant double complex
Suppose we have some (homological) double complex $\{E_{pq}\}$ with $p$ labelling the row and $q$ the column (is this standard or not?). Taking the homology of the vertical maps, it's easy enough to ...
0
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0answers
39 views
Endomorphisms of $\mathbb Z$-modules are the same as those of $\mathbb F_p$-vector spaces
Let $p$ be a prime number. Consider a direct sum of $n$ copies of the cyclic group of order $p$, written $C_p$: $G=C_p^{\oplus n}$. By a comment in this question, given an $R$-module $M$ and an ...
3
votes
2answers
29 views
$N/K$ is direct summand of $M/K$ implies $N$ is direct summand of $M$
Let $K\subset N\subset M$ be $R-$submodules where $R$ is a commutative ring with unity. If $K$ is a direct summand of $M$ then show that $K$ is a direct summand of $N$. Further, if $N/K$ is a direct ...
2
votes
1answer
33 views
Cartesian product of fields is semisimple iff the index set is finite
This question may have already been asked.
Let $R = \prod_{i\in I} K_i$ where each $K_i$ is a field. Show that
$R$ is a semisimple ring iff the index set $I$ is finite.
I think I need to show ...
1
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1answer
17 views
Direct sums of left ideals and orthogonal idempotents
Let $R = L_1 \oplus L_2 \oplus ··· \oplus L_n$ where each $L_i$ is a
left ideal and $n \in \mathbb{N}$. Show that there exist idempotents
$e_i \in R$ where $e_i = e_i^2$ such that
$1_R = \sum\...
2
votes
1answer
57 views
Confusion about the quotient $\mathbb Z^4/H$
Let $f:\mathbb Z^3\to\mathbb Z^4$ be the group homomorphism given by $$f(a,b,c)=(a+b+c,a+3b+c,a+b+5c,4a+8b).$$ Let $H$ be the image of $f$. Find an element of infinite order in $\mathbb Z^4 /H$ and ...
3
votes
1answer
60 views
Projective and flat vs. faithfully flat
Let $R$ be a commutative ring with unity and let $M$ be a projective and faithful $R$-module. Then is $M$ faithfully flat ? Is it true at least if $M$ is finitely generated, or say Noetherian ?
I ...
1
vote
1answer
17 views
Nilpotent Jacobson radical
I have to prove the following: If $R$ is a finite-dimensional algebra over a field $F$, then $J(R)$ is nilpotent.
I thought about this, but there are some gaps:
Because $R$ is in particular a ...
1
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1answer
23 views
Bases and generating list of $\mathbb Z/6\mathbb Z$ as $\mathbb Z$ module.
I need to find all free list, generating list and bases of $\mathbb Z/6\mathbb Z$ as $\mathbb Z/6\mathbb Z-$module and as $\mathbb Z-$module.
As $\mathbb Z/6\mathbb Z-$module : $\{1\}$ and $\{5\}$ ...
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2answers
27 views
Show that $(X,X^2+1)$ generate $A[X]$ as $A[X]$-module but is not free.
Let $A$ a commutative ring. Show that $(X,X^2+1)$ generate $A[X]$ as $A[X]$-module but is not free.
For the "generating" I tried by induction : if $a\in A$, then $$a=(X^2+1)-aX(X).$$
Then suppose it'...
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votes
0answers
31 views
quotient module with dimensional 1
If $V=\bigoplus V_{i}$ is an n-dimensional $R$-module and $W$ is a hyperplane "submodule of V" whose coordinates sum equal to zero (thus it is of $\ n-1$ dimension). If the quotient space $V/W$ is ...
1
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1answer
24 views
characterization of submodule of and of linear application.
Let $A$ a commutative ring, $G$ an abelian group and $K$ a field.
1) Characterize submodules of $A$ for its natural structure of $A$-module
2) Characterize submodules of $G$ for its natural ...
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0answers
40 views
What does it mean for $\mathcal{C}$ to be a $\mathcal{D}$-module, when $\mathcal{C}$ and $\mathcal{D}$ are categories?
If $\mathcal{C}$ is a module over a category $\mathcal{D}$, what does this mean? I looked around on line but couldn't find a definition.
My guess is that this means there is a functor $\mathcal{D}\to\...
2
votes
0answers
53 views
Grassmannian as a functor and the rank of a projective module.
Consider the following definition, taken from Introduction to Algebraic Geometry and Algebraic Groups by Demazure and Gabriel:
3.4 Example: Let $n,r$ be two integers $\geq 0$; the Grassmannian is ...
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2answers
39 views
What is a free module
Here I asked a question about the tensor products- Construction of tensor product over module and apparently I do not understand what a free module is.
More specifically, in page 24 of Atiyah's ...
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0answers
38 views
Union of modules indexed by ordinals
This was in pg103, Intro to Homological Alg, by JJ Rotman.
The same reasnoing applies to an ascending transfinite sequence of submodules $(P_\alpha)$ indexed by some well ordered set. The reader ...
0
votes
1answer
26 views
Free module over a group transfers to a free module over a subgroup
Let $H$ be a subgroup of $G$.
1- Why is $\mathbb{Z}G$ a free module over $\mathbb{Z}H$?
2- Why does every free $\mathbb{Z}G$-module is also free $\mathbb{Z}H$-module?
2
votes
1answer
76 views
How does linear algebra over the octonions and other division algebras work?
An interesting question, which has been discussed in many forms on this site, is how many results from the study of linear algebra over vector spaces carries over when we allow the scalars to form an ...
5
votes
1answer
59 views
Why is the isomorphism of Artin-Wedderburn a ring isomorphism?
I'm somewhat confused on the following calculation in the proof of Artin-Wedderburn: let $R$ be a semisimple ring such that (since $R$ is f.g. over itself) we have $R \simeq \oplus_{i = 1}^r S_i^{n_i}$...
0
votes
0answers
20 views
On isomorphism of quotient groups of free abelian groups of finite rank
Consider the free abelian group $\mathbb Z^n$, with elements considered as row vectors. For every $A\in M_{r\times n}(\mathbb Z)$ , let $K_A$ be the subgroup of $\mathbb Z^n$ generated by the row ...
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0answers
52 views
If $R^n$ is generated by $\{e_1, \cdots e_m\}$ as an $R$-module, then $n\leq m$
Given a commutative ring $R$, I want to show
If $R^n$ is generated by $\{e_1, \cdots e_m\}$ as an $R$-module, then $m \geq n$.
This follows from the standard result that if there exists a ...
3
votes
2answers
70 views
Making sense of defining tensor products $\bigotimes V$
This question is making sense of a definition. Suppose $R$ is a commutative ring.
How does one make a meaningful definition of $\bigotimes_1^k V_i$ where $V_i$ are $R$-modules? I know of the $k-$...
