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

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5
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1answer
150 views

What is $\operatorname{Hom}_R(P,R)$ isomorphic to when $P$ is projective?

Let $R$ be a (possibly noncommutative) ring with $1$. Now, quite clearly we have $$\operatorname{Hom}_R(R^n,R)\cong R^n.$$ I am wondering if there is any similar result for ...
4
votes
1answer
64 views

Two modules are isomorphic in the stable module category iff they are projectively equivalent

Let $R$ be a (not necessarily commutative) ring. Let ${\text{mod-}R}$ be the category of finitely generated right $R$-modules. Let $\underline{\text{mod-}R}$ be the stable module category, with the ...
2
votes
1answer
33 views

why is a simple ring not semisimple?

A simple module is a semisimple module . A module $M$ is called semisimple if every submodule is a direct summand of $M$ Since a simple module has $\{0\}$ and $M$ as its submodules so it is ...
1
vote
2answers
22 views

Do modules have to be defined over rings with unity?

This is definition for left module over a ring $R$ given in Wikipedia: Suppose that $R$ is a ring and $1_R$ is its multiplicative identity. A left $R$-module $M$ consists of an abelian group ...
1
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1answer
25 views

A question from the proof of: If M is a free f.g R-module, and N is a submodule of M, then N is free, where R is a PID

I'm reading the proof from Dummit and Foote third edition, page 460-461,theorem 4. I am confused about one thing: On page 461, there are: (a) $M=Ry_1\oplus \ker v$ (b)$N=Ra_1y_1\oplus (\ker v\cap ...
0
votes
1answer
18 views

Projective space of a module

I'm studying projective geometry in a basic course of geometry. My question is: Is there an equivalent definition of projective space not of a vector space but of a module? I think the basic ...
3
votes
2answers
52 views

relation of R-module homomorphisms with direct sums

If $f$ is a module homomorphism from $A$ to $B$ and $g$ is a module homomorphism from $B$ to $A$, and $g \circ f$ is the identity function, why is it that $B= \operatorname{im}(f) + \ker(g)$?
2
votes
1answer
25 views

Finitely generated and free module

We work in the category $R-\mathsf{Mod}$ where $R$ is unital. Some authors define free modules to have a finite basis. If we don't require a basis to be finite, I think it is quite obvious that a ...
0
votes
1answer
37 views

Every projective $R$-module $P$ is free

I have come across a theorem which states that if the underlying ring $R$ is a principal ideal domain then every $R$-module $P$ which is projective is free also. But the problem is I have encountered ...
0
votes
0answers
46 views

How bad must be a ring to allow cyclic artinian modules that are not noetherian?

I've been studying the relations between artinian and noetherian modules over commutative rings. One can prove two interesting results for the commutative case. Theorem Every commutative artinian ...
2
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0answers
126 views

Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
0
votes
1answer
26 views

Inverting a nonzerodivisor of a module

I'm reading the Paper "What makes a complex exact?" by Eisenbud and Buchsbaum. On page 266 it says: Thus we may assume $0 \neq \operatorname{rank}(\phi_n,L) < \operatorname{rank}(F_n)$ and ...
-1
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3answers
38 views

Is there an easy example of a module which is not projective

Is there an easy example of a module which is not projective? I found that direct product of $\mathbb Z$ is not projective but its proof is complicated. Are there any easy examples for this?
2
votes
2answers
69 views

Characterization of projective modules?

I'm having a hard time with a characterization of projective modules: A $R$-module $P$ is projective if and only if for every epimorphism $f:I\longrightarrow I^{\prime\prime}$ with $I$ injective and ...
2
votes
0answers
21 views

How to show that direct product of $\mathbb Z\times\mathbb Z\times\mathbb Z\times…$ is not projective as a $\mathbb Z$ module? [duplicate]

How to show that direct product of $\mathbb Z\times\mathbb Z\times\mathbb Z\times...$ is not projective as a $\mathbb Z$ module? I know that $\mathbb Z$ is a free $\mathbb Z$ module since it has ...
5
votes
0answers
24 views

Why not take the tensor product of two left modules in this way? [duplicate]

Let $A,B$ be two left $R$-modules. I was wondering if we then can form the tensor product of $A$ and $B$ by the free abelian group on $A \times B$ divided out by the span of the following elements ...
-2
votes
1answer
34 views

Quotient module and submodule.

Set $M$ as an $R$-module, where $R$ is a PID, then if any quotient module of M can be seen as a submodule of M? It seems not always true, and could you give me a counterexample? Thank you!
0
votes
1answer
30 views

If $M=\langle v_1,…,v_n\rangle $ is an $R$-module, is $\langle v_1\rangle$ always a submodule of $M$?

Note $R$ is a PID. In my mind, as $\langle v_1\rangle$ is a subgroup of $M$, but maybe it's not closed under scalar multiplication. Could you give me an counterexample, if $\langle v_1\rangle $ is ...
0
votes
2answers
47 views

Proof about finitely generated torsion-free R-module M is free, where R is a PID

Proof about finitely generated torsion-free $R$-module $M$ is free, where $R$ is a PID. Can I prove by the following way? Proof by induction on $n$, where $M=\langle v_1,...,v_n\rangle$. If ...
0
votes
1answer
40 views

Finite submodules of infinite torsion module

Let $R$ be a commutative ring and let $M$ be an infinite torsion $R$-module. Does $M$ have an infinite sequence of finite submodules of increasing cardinality? If not, is it true given additional ...
1
vote
3answers
105 views

All simple modules are projective $\Rightarrow$ semisimple [duplicate]

Let $A$ be a finite dimensional algebra over a field $K$. It is clear that if $A$ is semisimple, then every simple module is projective. Does the converse hold ? It seems false, but I can't find a ...
2
votes
1answer
36 views

Computing injective hulls over a lower triangular matrix ring

Let $R$ be the ring $\begin{pmatrix} {\mathbb Z }_{ p } & 0 \\ {\mathbb Z }_{ p } & {\mathbb Z }_{ p } \end{pmatrix}$ with $p$ prime. $R$ is an artinian ring and is also a $\mathbb ...
6
votes
1answer
130 views

Prove $AB=I$ implies $BA=I$ using Fitting's Lemma

I know this question has already been asked, but I need a proof that for $A,B \in M_n(K)$, $$AB=I_n \Rightarrow BA=I_n$$ using Fitting's lemma. I thought of using the fact that $K^n$ is a ...
4
votes
1answer
53 views

Right modules Vs Left modules.

I have been reading Frobenius Algebras, Volume 1 By Andrzej Skowroński, Kunio Yamagata. On page 18 I came across the following paragraph, and I founded interesting, I will quote it and then ask my ...
1
vote
2answers
22 views

Proving that an $FG$-homomorphism is surjective

Assume that $V$ is an $FG$-module.Prove that the subset $$V_0 = \{v \in V : vg = v \space \forall \space g \in G \}$$ is an $FG$- submodule of $V$. Also show that the function $$\phi: v \to ...
1
vote
0answers
39 views

is $Hom(P,N \otimes_{End(P)} P) = N$?

This is probably well known to people who work with algebras but I couldn't find a reference. Say I have a ring A and a module P and I take B = End(P), the endomorphism ring. Let N be a B-module, is ...
0
votes
1answer
21 views

Diagram with exact sequences of modules

I am trying to solve these two problems on diagrams of module morphisms: 1) Let $$\begin{array}{c} M' & \xrightarrow{f_1} & M & \xrightarrow{f_2} & M'' \\ & & ...
4
votes
0answers
53 views

Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.

See image below. I just want help proving that all kernels in $R$-Mod are monics. My attempt: Let $f : A \to B$ be a map in $R$-Mod. Suppose $i$ is a kernel of $f$, that is: $fi = 0$ and ...
0
votes
1answer
31 views

Elements of $M\otimes_{\mathbb K}\mathbb K[t]$?

I'm studying tensor product of modules over algebras and I came across the following statement. Given a $\mathbb K$-module $M$, the elements of $M\otimes_{\mathbb K}\mathbb K[t]$ can be written as ...
0
votes
1answer
8 views

Show that any two elements of $M$ are linearly dependent over $\mathbb Z$.

Let $M$ be the additive group of rationals .Show that any two elements of $M$ are linearly dependent over $\mathbb Z$. I do not understand why is this true.If $c_1\dfrac{a}{b}+c_2\dfrac{c}{d}=0$ then ...
2
votes
0answers
32 views

homology commutes with direct product of chain complexes. Direct proof

This is an attempt to prove that direct product of chain complexes commutes with homology (exercise in Weibel's book). I've had some success since I've proved that $Z_n(\prod_{\alpha \in A} C_{\alpha ...
0
votes
1answer
24 views

Weibel exercise 1.1.2. the $n$th homology module is a functor from category Ch-Mod$(R)$ to Mod-$R$

Ch-Mod$(R)$ is the category of $R$-module chain complexes. How do you turn a homology module into a functor? Thanks for teaching.
0
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0answers
20 views

Weibel's book exercise 1.1.2. Cycles get sent to cycles by chain complex homs $u : C_{\cdot} \to D_{\cdot}$

A morphism of chain complexes is a family of homs $u_n : C_n \to D_n $ such that $u_{n-1} d_n^{(C)} = d_n^{(D)} u_n$. Weibel's book says that cycles "get sent to cycles". To me that means that ...
1
vote
1answer
34 views

Is this a typo in Weibel, page 1?

It says a morphism $u : C_{\cdot } \to D_{\cdot}$ of chain complexes is a family of homomorphisms $u_n : C_n \to D_n$ such that $u_{n-1} d_n = d_{n-1} u_{n}$, but shouldn't it just be that $u_{n-1} ...
0
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0answers
32 views

How do I write a correct answer to Weibel exercise 1.1.1.?

Exercise 1.1.1. Set $C_n = \Bbb{Z}/8$ for $n \geq 0$ and $C_n = 0$ for $n \lt 0$. Let $d_n : x \pmod{8} \to 4x \pmod{8}$ Compute the homology modules of the chain complex $C_{\cdot}$. I got that ...
2
votes
2answers
60 views

Artinian iff finite dimensional

Let $F$ be a field and $M$ a finitely generated $F[x]$-module. Show that $M$ is Artinian iff the dimension of $M$ over $F$ is finite. Thoughts so far: For the backward direction, I assume for the ...
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0answers
21 views

$S$ subring of $R$. Is a projective objects in $R$-$\bmod$ still projective in $S$-$\bmod$?

Let $R$ be a ring (not necessarily commutative and not necessarily with unit). Recall the definition of $R$-$\bmod$ as an abelian group $A$ on which $R$ acts on the left respecting the following ...
3
votes
1answer
80 views

What is the relation between $C^\infty$-linear and tensorial?

I've been self studying Riemannian Geometry through Spivak's and Lee's books, and fairly often I've seen an argument that goes somewhat like this: We have an operator that acts on vectors in the ...
1
vote
2answers
19 views

Problem in the proof of whether two bases are equipotent or not of an $R$ module $M$

I am reading T.S.Blyth's Module Theory where it is written that two bases of an $R-$ module may have different cardinalities but if the ring $R$ is commutative then any two bases(if exist) of an $R$ ...
4
votes
1answer
34 views

Right projective but not bimodule projective

Let $R$ be a ring with $1$. What is an example of a right $R$-module $M$ (and a ring $R$) such that $M$ is projective as a right $R$-module but not projective as an $R$-bimodule (assuming $M$ has ...
3
votes
2answers
36 views

Torsion of $\mathbb Z ^{2 \times 2}$

I am trying to find the torsion of the ring $R= \mathbb Z ^{2 \times 2}$ seen as an $R-$ left module. So I want to see for which matrices $B$, there exists $A \neq 0$ such that $AB=0$. So if ...
0
votes
0answers
19 views

Unique module homomorphism

Let $R$ be an integral domain $M$ be $R$-module. $S^{-1}M$ is localization of $M$ at $S$ as the $S^{-1}R$-module $i_M:M\rightarrow S^{-1}M$ ; $m\mapsto (m,1)= m/1$ I showed that this is an ...
0
votes
1answer
32 views

Localization of module

Let denote localization of $M$ at $S$ as the $S^{-1}R$-module $S^{-1}M$ Question is as follows: Let $R$ be an integral domain and $M$ is an $R$-module. Show that $M=0$ iff $S^{-1}M=0$ for all ...
3
votes
1answer
43 views

Endomorphism ring of $x \Bbbk\langle x,y \rangle + y \Bbbk\langle x,y \rangle$

Let $R = \Bbbk \langle x,y \rangle$, where $\Bbbk$ is a field. I want to determine $\underline{\text{End}}_R(xR + yR)$, the ring of (not necessarily degree-preserving) graded module homomorphisms ...
1
vote
1answer
24 views

There exists $h:P\longrightarrow M$ such that $fh=g\Leftrightarrow \textrm{Im}(g)\subseteq \textrm{Im}(f)$?

Let $R$ be a ring with identity and $P$, $M$ and $N$ three left modules over $R$. Futhermore, suppose $P$ is projetive. Let $f\in \textrm{Hom}_R(M, N)$ and $g\in\textrm{Hom}_R(P, N)$. How can I show ...
1
vote
2answers
122 views

Is a specific ring extension $B$ of $K[x,y]$ integrally closed? separable?

Let $A=K[x,y] \subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
3
votes
1answer
89 views

Surjection $M\to R/P$

Let $(R,\mathfrak{m})$ be a local Noetherian ring, and $M$ a finitely generated $R$-module. I am trying to show that there is a surjection $M\to R/P$ for any $P\in\operatorname{Supp} M$. I know ...
6
votes
1answer
49 views

Need for projective modules

I wanted to ask why we require projective modules. After studying all the essential ingredients my guess is - Firstly, we worked with vector spaces (say modules over field $F$) (which are free ...
2
votes
0answers
30 views

How to write isomorphisms $A^* \otimes B \cong \hom(A, B)$ and $\hom(A\otimes B, C) \cong \hom(A, B^* \otimes C)$ explicitly?

I would like to write isomorphisms $A^* \otimes B \cong \hom(A, B)$ and $\hom(A\otimes B, C) \cong \hom(A, B^* \otimes C)$ explicitly. I think that $\varphi: A^* \otimes B \to \hom(A, B)$ is given by ...
2
votes
2answers
36 views

Conceptual doubt about Modules

Let $R$ be a ring. Then a left $R$- module is an additive Abelian group M along with an operation $R\times M \to M$ such that it satisfies - $r(x+y)=rx+ry$ $(r+s)x=rx+sx$ $(rs)x=r(sx)$ $1_R x=x$ ...