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

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33 views

Show that $D^n$ is simple and unique [duplicate]

Suppose $A=M_n(D)$, i.e. n by n matrices over a division algebra $D$. We want to prove that: $D^n$ (viewed as row vectors) is a simple right $A$-module and any other simple $A$-module is ...
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1answer
101 views

Doubt about a kernel

I was reading the proof of Lemma 1.25 in this thesis and I thought I understood it, but I think I don't. The thing that I don't see clearly is in page 26 where he is showing that $\textrm{ker}\ ...
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0answers
37 views

Showing that an element generates the kernel

$I $ is a monomial ideal generated by $\left < m_1, \dots, m_n\right >$ and suppose we also have an $R$-module homomorphism $\phi: \oplus_{j = 1}^n Re_j \to I$ defined by $$\phi(e_i) = m_i.$$ ...
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0answers
47 views

The $R/I$-module $F/IF$.

Let $F = \oplus_{i = 1}^r Re_i$ be a free $R$-module and $I\subset R$ be an ideal and $IF = \{ if :i \in I, f \in F \}$ is a submodule of $F$. Then (1) $F/IF$ is an $R/I$-module and (2) $\{e_i ...
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1answer
20 views

Proof of Quaternion Algebras being Division Algebra or $M_2(F)$

I have a question regarding the proof of quaternion algebras (denoted $A$) being either a division algebra or isomorphic to $M_2(F)$. I can understand that $A$ is central simple. And since $A$ is ...
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1answer
63 views

Show that a sequence is a free resolution

Let $I \subset R = k[x_1,\dots,x_n]$ be an ideal and $f \in R$ such that $I = \left < f \right >$ ($k$ is a field, so R is commutative ring). How do I show that (1) $I$ has a free resolution ...
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1answer
26 views

Is this a semisimple algebra?

Let $A$ be the set of 2 by 2 matrices of the form $\begin{pmatrix}d_1 &d_2\\d_3 & d_4\end{pmatrix}$, where $d_i$ are in division algebras $D_i$. Update: $D_1=Hom(N_1,N_1)$, ...
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1answer
28 views

Fiber Product diagram

I have the following problem that I can't seem to figure out the trick to. I have the following commutative diagram: $X=\{(a,b): a\in A, b\in B$ such that $f(a)=g(b)\}$ is the fiber product, f,g ...
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0answers
50 views

Proving $End_B(N)$ is Semisimple Algebra

Let $B$ be a semisimple algebra, and $N$ be a finitely generated $B$-module. I am trying to prove $End_B(N)$ is a semisimple algebra. Updated: I changed my attempt based on user26857's and Qiaochu's ...
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0answers
51 views

Flaw in this proof (direct sum of modules)

I am trying to prove the following: If $$\bigoplus_{i\in I}\alpha_iN_i\cong\bigoplus_{i\in I}\beta_iN_i$$ where $N_i$ are distinct simple modules, then $$\alpha_i=\beta_i$$ for all $i\in I$. It is ...
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1answer
32 views

Notation of plus with a dot above it \dotplus

Question regarding notation of internal/ external direct sum. I am referring to Richard Pierce's book Associative Algebras, where he uses two notations for direct sum $\oplus$ and $\dotplus$. I am ...
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3answers
46 views

Submodules of a module with a given property

I am curious about the submodules of a module with a given property. Let $M$ be an $R$-module. If $M$ is a finitely generated are the submodules of $M$ finitely generated? If $R=\mathbb Z$, $M$ ...
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1answer
23 views

Understanding semisimple Z modules

I'm trying to understand the nature of semisimplicity in Z modules. So for instance would I be right in thinking $Z/30Z \oplus Z/2Z$ is semi simple as it can be expressed as $Z/3Z \oplus Z/5Z \oplus ...
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2answers
60 views

Why isn't every $\mathcal O_X$-module quasi-coherent?

This might be a stupid question, but I don't understand an easy fact. Let $(X,\mathcal O_X)$ a ringed space. We know that every module $M$ over a ring $R$ has a free presentation, so why isn't every ...
2
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1answer
29 views

Module over a commutative ring with a topology

Let $M$ be an $R$-module ($R$ commutative ring with unity). Let $M=M_0 \supseteq M_1\supseteq M_2\supseteq\cdots$ be a chain of submodules. The topology in $M$: The open sets in $M$ are arbitrary ...
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1answer
24 views

if $e$ is an idempotent, then $Re$ is a projective module

I would like to show that if $R$ is a ring with $1$, and $e$ is an idempotent in $R$, then $Re$ is a projective module. My idea is to show that $R = Re \oplus R(1-e)$. It's clear to me that $R = Re ...
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1answer
13 views

Exponent of submodule is a divisor of the exponent of module

Let the exponent of an R-module be the generator of the group of annihilators. Let R be a PID and M be a torsion R-module with exponent $r \neq 0$. I want to show that any submodule $N ≤ M$ will have ...
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0answers
51 views

Are the following monomial modules?

If $I$ is an monomial ideal of $R$ and $M, N$ are monomial modules of $\oplus_{i = 1}^{r} Re_i$, then the following are monomial modules. Why? 1) $M + N = \{m + n: m \in M, n \in N \}$ (because it's ...
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1answer
20 views

How to find an $R$-order $\Lambda$, such that $A=k\Lambda$?

Let $R$ be a Noetherian integral domain with quotient field $k$. An $R$-order $\Lambda$ is a finitely generated torsionless non-zero $R$-module, which is at the same time an $R$-algebra. Why is ...
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1answer
35 views

If $\mathbb{C}[x]/(x^n)$ is an Artinian $\mathbb{C}$-module, is it a Artinian ring?

I feel I should use the fact that ideals of $\mathbb{C}[x]/(x^n)$ are equivalent to submodules $\mathbb{C}[x],$ but I cannot see how to use this fact to prove that $\mathbb{C}[x]/(x^n)$ is an Artinian ...
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1answer
34 views

If $A$ is a semisimple algebra, then every irreducible $A$-module is cyclic over $A$

Let $A$ be a finite-dimensinoal associative algebra with unit $1$ over $K$. Then $A$ is a $A$-right module (by module I always mean right module) over itself by right multiplication (the so called ...
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1answer
38 views

Is the category of left modules is isomorphic to the one of right modules?

Given a ring $R$, it is said in "Abstract and concrete categories" by Adamek et al. (p 25) that the category of left modules $R\textbf{-Mod}$ is isomorphic to the one of right modules ...
5
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1answer
120 views

Composition series for Verma modules.

Let $L$ a Lie Algebra. I need prove that that every Verma module $\Delta(\lambda)$ admits a composition series, i.e a series of submodules with simple factors. I found a proof that is quite short in ...
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1answer
53 views

$A$-linearity of the category of $A$-modules.

Let $A$ be a noncommutative ring. I asked myself whether the category of $A$-modules is an $A$-linear category or not. I am not able to find that this is true by giving the appropriate structure. Now ...
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1answer
21 views

Rings of finite uniform dimension

Let a ring $R$ has finite Goldie dimension, say, as left $R$-module. Could it be deduced that $R$ is $I$-finite in the sense that $R$ does not contain an infinite subset of non-zero orthogonal ...
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1answer
43 views

Can a module be viewed as a functor?

A monoid action can be viewed as a functor from $\textbf{M}$ to $\textbf{Set}$ which is constant if $\textbf{M}$ is the monoid. Knowing that a vector space is a group action of a ring $A$ on a group ...
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1answer
72 views

Relation of $G$-invariants and $g$ -invariants.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $H$ a Hopf algebra and $M$ an $H$-module. By definition, $m \in M$ is called invariant if $x.m=\epsilon(x)m$, $\forall x \in H$, where $\epsilon: H ...
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2answers
47 views

Why is $\mathbb{Q}$ not semisimple as a $\mathbb{Z}$ module?

Just want to ask: Why is $\mathbb{Q}$ not semisimple as a $\mathbb{Z}$-module? Here is my attempt: I know that simple $\mathbb{Z}$-modules are isomorphic to $\mathbb{Z}/p\mathbb{Z}$. So if ...
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3answers
48 views

Finite length $A$-algebras are finitely generated?

Let $k$ be a field and $M$ a module over a (associative, unital) finite-dimensional $k$-algebra $A$. The length of $M$ is the unique length of a composition series for $M$. How does $M$ having finite ...
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0answers
35 views

Finitely generated torsion $R$-module with trivial annihilator

Let $R$ be a commutative ring. I would like to know if there are any examples of a finitely generated torsion $R$-module with zero annihilator. I know that if $R$ is a domain, this is impossible. So ...
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2answers
24 views

$M=K\oplus I$ iff there exists an idempotent $\phi \in End_R(M)$

I have a ring R and an R-module M. I want to show that $M=K\oplus I$, where $K=Ker(\phi)$ and $I=Im(\phi)$ iff there exists an idempotent $\phi \in End_R(M)$ I am happy showing that if $\phi$ is ...
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0answers
39 views

Ring homomorphisms between modules?

I'm reading Lang's Algebra and there's a statement at the beginning of the third chapter that's confusing me: Let $A$ be a ring, and let $X, X'$ be $A$-modules. We denote by $\text{Hom}_A(X', X)$ ...
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1answer
25 views

When are minimal faithful modules over algebras unique?

Let $A$ be a unital associative algebra over $\mathbb{C}$. We say an $A$-module $V$ is "minimal faithful" if (i) $V$ is faithful and (ii) $V$ does not have a proper submodule that is faithful. First ...
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1answer
14 views

Show that a 1 dimensional subspace of a CG-module is a submodule.

Good evening, I am trying to finish the following proof from an old exercise sheet I have from my third year of university a few years ago. Let $G$ be a finite group and $V$ be a finite-dimensional ...
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1answer
30 views

Torsion elements with relatively prime orders

Let $x$ and $y$ be torsion elements in an R-module $M$ having orders $a$ and $b$. With $a,b \in R$, $a$ and $b$ are relatively prime, and $R$ is a PID. I want to show that $x + y$ has order $ab$. So ...
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1answer
26 views

Elements in a symmetric algebra

If $V$ is a vector space of dimension $n$, what do elements in Sym$^k(V)$ look like? I've seen the "mod out by" definition, but I am still unclear as to what this vector space looks like. I have read ...
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1answer
51 views

$\mathbb{R}$ as a $\mathbb{Z}$ module

Is $\mathbb{R}$ free as a $\mathbb{Z}$ module? If it is, it must be of infinite rank, I suspected that some set theory will be involved, but have no clue about this fact.
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1answer
28 views

Structure of Modules and Elementary Divisors Matrix

Find the structure of the module $\mathbb{Z}^{3}/N$ where $N$ is spanned by $(1,1,1),(6,3,2), (4,1,0)$. I checked that the above vectors are not linearly independent. I have generally learned ...
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4answers
59 views

Why does free imply torsion free?

I want to verify that if R is an integral domain and M is an R-module, that if M is free, M must also be torsion free. Where can I start with this? I feel like it is obvious but I can't see it. I am ...
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1answer
87 views

Show that $\text{card}\left(\frac{\mathbb Z^n}{f(\mathbb Z^n)}\right)=|\det(M)|$

Let $f:\mathbb Z^n \to \mathbb Z^n$ be a group homomorphism represented with respect to the standard matrix $M\in M_n(\mathbb Z)$, and assume that $\det(M)$ isn't $0$. Show then that ...
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1answer
22 views

CG-homorphism proof. Stuck at the end!

I am trying to work on some questions back from my uni days, and one has gotten the better of me at the moment! Let $G$ be a finite group and $V, W$ finite-dimensional $\mathbb{C}G$-modules. Let ...
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2answers
29 views

Cosets of submodule

Suppose we have a ring $R$ and an ideal $J \subseteq R$. Let $M$ be an $R$-module and $N$ be a submodule of $M$. Assume for two elements $m$ and $m'$ of $M$ we have $$m+ JM=m'+JM.$$ Since $N ...
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0answers
81 views

Elementary divisors for chains of submodules

Given free modules $N \le M$ of finite rank over a PID $R$, it's well-known that there is a basis $\{x_1,\ldots,x_n\}$ of $M$ and there are $e_1,\ldots,e_n \in R$ such that $\{e_ix_i\mid e_i \neq ...
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1answer
36 views

A $R$-module over a ring $R$ with identity is free iff it is a direct sum of copies of $R_R$, where $R_R$ is $R$ considered as a module over itself

Let $R$ be a ring with identity and $M$ be an $R$-right module. Then $M$ is called free over $X \subseteq M$ if for every module $N$ with mapping $\alpha : X \to N$ we can extend it uniquely to a ...
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2answers
42 views

Generators for a free submodule of a free module

Suppose we are given a finitely generated free module $M$ over a ring $R$. Assume $N \subseteq M$ is a free submodule of $M$. If $B$ is a basis for $M$, does it follow that there exists a subset $A ...
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3answers
160 views

nilpotent endomorphism on finitely generated modules over a domain

If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$. The cases $ m \le n$ is trivial. When $m>n$, I don't have much ...
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0answers
25 views

Uniqueness submodule of cyclic module

I'm having some problems with an exercise from Hungerford's Book of Algebra. It states: Let $R$ be a PID, and $A$ a unitary $R$-module such that $A$ is cyclic of order $r$. a) Prove that ...
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2answers
31 views

CG-modules: what does this notation mean?

I am trying to solve a question, but I do not know what the notation used means. If anyone could help me out that'd be great! I don't need help doing the proof, just what the notation means would be ...
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1answer
23 views

Kernel of $M\to M[U^{-1}]$ and primary decomposition of $(0)$

I am working on exercise 3.12 from Eisenbud's Commutative Algebra and I am having trouble parsing the question. Let $M$ be a finitely generated module over the Noetherian ring $R$. Given any ...
1
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1answer
81 views

$\mathbb C$-dimension of vector space $\mathbb C\otimes_{\mathbb R}\mathbb C$

Let $\mathbb R$ be the field of real numbers, $\mathbb C$ be the field of complex numbers. Consider $\mathbb C\otimes_{\mathbb R}\mathbb C$ as a $\mathbb C$-vector space via $a(b\otimes c) := ab ...