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

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2
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0answers
36 views

Module isomorphisms and coordinates modulo $p^n$

Let $p$ be a prime and $n \in \mathbb{N}$ is such that $p^n > 2$. We let $\alpha \in \mathbb{N}$ be such that $0 < \alpha < n$. Let $R := \mathbb{Z}_{p^n}$ denote the ring of integers modulo ...
3
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2answers
54 views

Can we construct a homomorphism from a projective module into a free module?

In short, I have a projective module and a free module, and want to construct a module homomorphism between the two. Is this always possible, at least in some way? Let me go into more detail. Suppose ...
0
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0answers
18 views

Regarding Adjugate of a Module Homomorphism

$\newcommand{\adj}{\text{adj}}$ I am trying to understand the concept of adjugate of an endomorphism of free module of finite rank as discussed here in Section 8. Let $M$ be a free module over a ring ...
1
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1answer
40 views

Let $R$ be a commutative domain with field of fractions $F$. Prove that $F$ is an injective $R$-module.

Let $R$ be a commutative domain with field of fractions $F$. Prove that $F$ is an injective $R$ module. I have tried to apply Baer's criterion: every $R$-module homomorphism from any ideal $I$ of ...
1
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1answer
30 views

Prove that if $M$ is a simple $R=k[x_1,…,x_m]$ -module, then the dimension of $M$ over $k$ is finite.

Let $k$ be a field and let $R=k[x_1,...,x_m]$ be the polynomial ring in $m$ indeterminates. Prove that if $M$ is a simple $R$-module, then the dimension of $M$ over $k$ is finite. I think since ...
4
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0answers
30 views

Integer such that there is a $k$-algebra isomorphism for any two algebras.

Is there an integer $\ell = \ell(m, n) \ge 1$ such that for any $k$-algebras $A$ and $B$ there is a $k$-algebra isomorphism $\text{M}_m(A) \otimes_k \text{M}_n(B) \cong \text{M}_\ell(A \otimes_k B)$?
1
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1answer
26 views

Defining the dual-comodule of a comodule?

As is well known, every left module has a dual, which is a right module. How does this work for comodules? More explicitly, does there exist a notion of the ...
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0answers
13 views

Can We Use Extension of Scalars (Or Some Other Neat Way) to Prove This

Let $k$ be a field and $b(x)$ be a polynomial of degree at least $1$ in $k[x]$. Then for any given $f(x)\in k[x]$, there is an integer $m$ and polynomials $d_0(x), \ldots, d_m(x)\in k[x]$ such that ...
2
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1answer
55 views

The direct limit of morphisms and the direct limit of tensor product functors

While reading these notes I had something of an existential crisis, after realizing that my understanding of direct limits might somehow be fundamentally insufficient. In particular, alarms started ...
0
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0answers
20 views

Identifying targets of epimorphisms as cokernels of some morphism

In Aluffi's Algebra: Chapter 0 text, on P. 167, when discussing the category of R-Mod, he writes that "... every epimorphism identifies its target with the cokernel of some morphism." For any ...
4
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1answer
56 views

Showing that localization is an exact functor

I'm again in this awfully familiar situation where I'm struggling to prove simple statements mostly because I have no idea how a template of a proof should look like in this specified context. I'm ...
3
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0answers
28 views

Followup to my previous question, $M = \text{Image}(u^\infty) \oplus \text{Ker}(u^\infty)$.

See my previous question here, Intersection of images and union of kernels. Let $A$ be a ring (not necessarily commutative), let $M$ be an $A$-module, and let $u: M \to M$ be an $A$-module ...
2
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0answers
27 views

Ascending Sequence of Submodules

Claim: Consider an ascending sequence of submodules of a module $P$: $$\{0\} = P_0 \subseteq P_1 \subseteq P_2 \subseteq \cdots$$ where $P = \bigcup_{n \geq 0} P_n$. Suppose that $P_n$ is a direct ...
3
votes
2answers
25 views

Intersection of images and union of kernels.

Let $A$ be a ring (not necessarily commutative), let $M$ be an $A$-module, and let $u: M \to M$ be an $A$-module morphism. Put $$\text{Image}(u^\infty) := \bigcap_{k=1}^\infty \text{Image}(u^k),\text{ ...
3
votes
1answer
74 views

Step in five-lemma's proof.

Consider the following commutative diagram, where rows are exact: $$\require{AMScd} \begin{CD} M_1 @>f_1>> M_2 @>f_2>> M_3 @>f_3>> M_4 @>f_4>> M_5 \\ @Vh_1VV ...
1
vote
2answers
52 views

If $f\otimes_\mathbb{Z}\mathbb{Z}/(p)\colon M\otimes_{\mathbb{Z}}\mathbb{Z}/(p)\to N\otimes_\mathbb{Z} \mathbb{Z}/(p)$ is onto for all $p$, $f$ onto?

This lemma is used in a theorem I'm reading, with no proof. Suppose $f\colon M\to N$ is a morphism of free, finitely generated $\mathbb{Z}$-modules. Then if $f\otimes_\mathbb{Z}\mathbb{Z}/(p)$ is ...
3
votes
2answers
80 views

Paul Garret's Proof of the Cayley-Hamilton Theorem

I am trying to understand the proof of Cayley-Hamilton Theorem given in Paul Garrett's notes. We have a finite dimensional vector space $V$ over a field $k$ and are given a linear operator $T\in ...
1
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0answers
57 views

Constant Dimension for Localization of Projective Modules

It is a well known fact that the localization of a projective module over a commutative ring is free. However, I don't know anything about the dynamics of how the dimension of the resultant free ...
0
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1answer
19 views

A question on $R$-modules

Let $M$ be a non-trivial irreducible (simple) $R$-module . Let $0 \ne m \in M$ and $A(m_0):=\{x \in R: xm_0=0\}$ , then is $A(m_0)$ a maximal left-ideal of $R$ and as $R$-modules , $M$ and $R/A(m_0)$ ...
1
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1answer
18 views

Let $M$ be a non-trivial $R$-module having no non-trivial submodule; then every non-trivial $R$-homomorphism on $M$ is isomorphism?

Let $M$ be an irreducible $R$-module such that $\exists r_0 \in R$, $\exists m_0 \in M$ such that $r_0m_0 \ne 0$. Then how to prove that any $R$-homomorphism $T$ of $M$ into $M$ is either an ...
1
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1answer
42 views

dual algebra of a coalgebra

Given a coalgebra $A$ over a field $F$ (for example, $H_*(X:F)$, i.e. the homology of a space $X$ equipped with $\Delta_*$, where $\Delta: X\to X\times X$, $x\to (x,x)$) how to obtain its dual $A^*$ ...
0
votes
1answer
31 views

Submodules of Noetherian modules

$k(x_i)_{i \in \mathbb{N}}$ is a $k[x_i]_{i\in \mathbb{N}}$-module. $k[x_i]_{i\in \mathbb{N}}$ is a $k[x_i]_{i\in \mathbb{N}}$- submodule of $k(x_i)_{i \in \mathbb{N}}$. $k(x_i)_{i \in \mathbb{N}}$ ...
0
votes
1answer
37 views

Proving that this function is an Endomorphism?

Given that $\mathbb{C}G = W_1 \oplus W_2$ and $1 = e_1 + e_2$ where $e_1 \in W_1$ and $e_2 \in W_2$ Also Knowing that $$w_1e_1 = w_1, w_2e_1 = 0$$ $$w_1e_2 = 0 , w_2e_2 = w_2$$ Let $x \in G$ Now it ...
1
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1answer
37 views

Injectivity, Projectivity, and $P$-injectivity of Localization [closed]

Let $R$ be a commutative ring with unity. I have read that if $M$ is an injective $R$-module, then $S^{-1}M$ is not necessarily an injective $S^{-1}R$-module. I need an example... Does last ...
2
votes
1answer
61 views

Multilinear Algebra Proof of the Cayley-Hamilton Theorem.

I am trying to understand the proof of the Cayley-Hamilton Theorem given in Section 4 of this document. On pg. 4 of the document, there is a line which reads: From general multilinear algebra, we ...
2
votes
1answer
56 views

Does tensoring with a finite-dimensional module preserve products?

Let $R$ be any $k$-algebra, $k$ a field. Suppose $M$ is a finite dimensional $R$-module. When does $M \otimes_{R} - $ commutes with arbitrary direct products? Any hints on why this is true? Is there ...
1
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1answer
28 views

To Show that $R/I\otimes_R R/J\cong R/(I+J)$

Let $R$ be a commutative ring with identity and $I$ and $J$ be ideals in $R$. Show that $R/I\otimes_R R/J\cong R/(I+J)$ as $R$-modules. This is what I tried. Define a map $f:R/I\times R/J\to ...
4
votes
1answer
46 views

Gauss sums and module endomorphisms

Let $p$ be an odd prime and $n \in \mathbb{N}$. Let $a,b,c$ be arbitrary integers such that $ab \neq 0$. We write $p^{\alpha}A = a$ and $p^{\beta}B = B$ for some $\alpha, \beta \in \mathbb{N}_0$ and ...
2
votes
1answer
42 views

Which primes belong to the support of $S^{-1}M$ with respect to $R$

Let $R$ be a noetherian ring, $S$ a multiplicatively closed subset of $R$ and $M$ a finitely generated $R$-module. The support of $M$, denoted ${\rm Supp}_R(M)$, is the set of prime ideals $p$ of $R$ ...
0
votes
1answer
65 views

Isomorphism theorem for modules

Let $p$ be a prime and $n$ be a positive integer such that $p^n > 2$. Let $R:= \mathbb{Z}/p^n \mathbb{Z}$ and suppose $(x,y) \in R \times R$, where addition and multiplication are defined ...
3
votes
0answers
31 views

Does $M \otimes_R N = 0$ for a non-unital ring $R$ if there are ideals $I,J \lhd R$ such that $MI+JN = 0$ and $I+J = R$?

Suppose that $M,N$ are $R$-modules, $I,J\lhd R$. Suppose that $MI=JN=0$ and that $I+J=R$. Prove that $M\bigotimes_R N=0$. This is very easy to prove if the ring is unital as you may write ...
4
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0answers
103 views

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} & ...
1
vote
1answer
45 views

Localization of a coherent module is coherent

I would like to prove that the coherence of a sheaf on a scheme is affine-local. For this, it is necessary to prove that for a coherent $A$-module $M$, its localization $M_{f}$ at $f\in A$ is a ...
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0answers
44 views

Example of two modules M, N where the set of the $m\otimes n$ is not a submodule

If I remember my Algebra class correctly, this is in general not true because, for example, the addition isn't closed. So if I take: $B:\mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}^{2^2}\cong ...
2
votes
1answer
39 views

A simple divisible module

Let $k$ be division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\text{End}_k(V)$ be the ring of right linear transformations on $V$. My questions are: (1) Is $V$ a simple ...
4
votes
1answer
38 views

$Z_n$ as $Z[i]$-module

I am trying to find all $n$ such that $Z_n$ is a $Z[i]$-module, where $Z[i]$ is a Gaussian integer ring. I proved that any $Z[i]$-module $M$ is just an Abelian group with hommomrphism ...
0
votes
0answers
32 views

Under what conditions is a ZG-module torsion-free?

If we have a ZG-module A, I was wondering if there are known condition we may imply on either A or the group G to make A torsion-free?
2
votes
1answer
29 views

Difference between torsion and zero divisor

I'm not understanding what the difference is between a zero divisor and a torsion element of a module. My best guess is that the torsion elements are "vectors" and zero divisors are scalars. This ...
5
votes
1answer
74 views

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module. I have tried this one and got $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
0
votes
1answer
47 views

What's a diagonal sum of two matrices?

Let $A$ be an $n\times n$ matrix over a field $K$. Show that there exists an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal sum of an invertible matrix and a nilpotent matrix. (Hint: use ...
4
votes
2answers
69 views

“If $x$ is a non-unit, then $1-ux$ is a unit”

I don't understand these two lines from my book. We are given that $R$ is a local ring. If every $2$-generator submodule is cyclic and $Ra$, $Rb$ are given, then $Ra+Rb=Rc$, hence ...
0
votes
1answer
35 views

What's a $2$-generator submodule?

Context: Let $R$ be a local ring and $M$ an $R$-module. Show that the set of all submodules of $M$ is totally ordered by inclusion iff every finitely generated submodule of $M$ is cyclic or, ...
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vote
3answers
78 views

What does $\overline{r}m:=rm$ mean?

On this Wikipedia article, it says that you can define an $R$-module $M$ as an $R/Ann_R(M)$-module using the action $\overline{r}m:=rm.$ What does that action actually mean? What is $\overline{r}$?
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0answers
22 views

Decompose finitely generated modules and use Krull-Schmidt theorem [duplicate]

I'm trying to show that if $R$ is an Artinian ring, then for finitely generated modules $M,N,N'$, we have that $M\oplus N\cong M\oplus N'$ implies that $N\cong N'$. I'm supposed to do this by ...
3
votes
2answers
159 views

Modules that have finitely many submodules

Drawing the lattice of submodules of a given module helps me to gain some intuition about the structure of module. Sometimes, however, it is not possible to draw in neat manner; For example vector ...
1
vote
1answer
46 views

$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that ...
0
votes
0answers
31 views

Free and projective modules

I have two simple questions. Let $R$ be a ring. Is every free $R$-Mod of the form $R^{(S)}$ for some set $S$? Can $R^S$ be projective for an infinite set $S$, for suitable, non-trivial $R$? Note that ...
4
votes
1answer
102 views

When $N \to M \otimes_R N$ is not an embedding.

Can someone please provide an example of the following (or tell me why such an example doesn't exist): Let $R$ be a (not necessarily commutative) ring, $M$ an $R$-$R$-bimodule containing a copy of ...
1
vote
1answer
30 views

Simple bimodule over a basic algebra

I am looking for a reference for the following result: Let $k$ be an algebraically closed field and $A$ be a finite dimensional $k$-algebra. If $A$ is basic, then every simple $A$-$A$ bimodule is ...
1
vote
1answer
31 views

How can I show $I\simeq L$ where $L$ is a direct summand of $X$ such that $L\subseteq \textrm{ker}(g)$?

Consider the exact sequence of $R$-modules ($R$ is ring with identity $1_R$): $$ 0\longrightarrow K\oplus I\stackrel{f}{\longrightarrow} X\stackrel{g}{\longrightarrow} Y\longrightarrow 0,$$ where $I$ ...