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Questions tagged [modules]

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

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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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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 \...
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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 $...
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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
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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 ...
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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 ...
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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 ...
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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!
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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 $\...
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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 $p(x) \in F[x]$ ...
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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: ...
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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 ...
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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?
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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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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},...
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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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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 $\mathfrak m$....
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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 ...
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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 ...
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1answer
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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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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 ...
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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 ...
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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 ...
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2answers
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$N/K$ 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 ...
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1answer
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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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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
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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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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 ...
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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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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\...
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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
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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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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 ...
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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?
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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 ...
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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 $R \simeq \oplus_{i = 1}^r S_i^{n_i}$...
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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 $k-$...