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

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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 ...
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1answer
27 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 ...
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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 ...
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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 ...
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2answers
21 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 ...
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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 ...
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1answer
20 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'' \\ & & ...
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51 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 ...
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1answer
30 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 ...
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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 ...
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0answers
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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 ...
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1answer
20 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.
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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 ...
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1answer
32 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} ...
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25 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 ...
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2answers
58 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
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$S$ 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 ...
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1answer
77 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 ...
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2answers
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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 $R$ ...
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22 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 ...
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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 ...
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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 ...
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1answer
29 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 ...
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1answer
41 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 ...
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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 ...
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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 ...
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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 ...
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1answer
47 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 ...
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0answers
27 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 ...
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2answers
35 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 - $r(x+y)=rx+ry$ $(r+s)x=rx+sx$ $(rs)x=r(sx)$ $1_R x=x$ ...
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2answers
29 views

Action of the endomorphism ring of a module

Let $A$ be a ring with $1$ and $M$ a left $A$-module. In chapter 6 of Curtis and Reiner's 'Methods of Representation theory', $M$ is regarded as a right module over the endomorphism ring ...
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1answer
37 views

Is $K[[x]]$ an Artinian/Noetherian $K[x]$-module?

Let $K$ be a field an consider $K[[x]]$ as a $K[x]$-module. Determine if it is Artinian/Noetherian. I used the following propositions: If M is an $R$-module and $N\subseteq M$ a submodule, then ...
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1answer
20 views

If $ R = L \oplus N $, then $ L = Re$ for idempotent $ e \in R $

I'm trying to prove the following statement. Suppose $ L \lhd R$ is a left ideal in $ R $ (a ring with unity). If there exists a left ideal $ N \lhd R $ such that $ R = L \oplus N $, then $ L = Re $ ...
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1answer
34 views

A question on modules of rings

Before arising my question, let us see the following fact. Let $R$ be a ring with unity and $R'$ be a subring of $R$. If $M'$ is a simple $R'$-module, there exists a simple $R$-module $M$ such that ...
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2answers
173 views

What exactly is a trivial module?

Yes, this is a quite basic answer, but I have to admit to be absolutely confused about this notion. Searching on the web, I managed to found two possible definition of trivial modules, referring ...
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0answers
33 views

Relation between Extensions and self-Extensions!

This should be considered as very general question regarding the extension group $Ext^i _A (R,S)$, in particular where $i=1$, for $R$ and $S$, a pair of given objects in an abelian category $A$. For ...
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0answers
13 views

Indecomposables generate the algebra

The following result seems intuitive, but I'm having hard time proving it, or determining if it is, in fact, false. Let $(A, \mu, \eta, \epsilon)$ be an augmented algebra, with augmentation ideal ...
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1answer
86 views

Homotopy equivalence and chain complexes

This is from the book: Hilton and Stammbach, A Course in Homological Algebra, Chapter IV, Derived Functors, exercise 4.2. Let $\varphi:C \to D$ be a chain map of the projective complex $C$ into ...
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2answers
39 views

Simple K[G] module has dimension 1 over K

Let G be a finite abelian group, K an algebraically closed field, K[G] the group algebra of G over K and M a K[G] module. I would like to show that if M is simple, it has dimension 1 over K. I think ...
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33 views

Row-Column Expansion

I saw a Proposition as follows: Let $A=(a_{ij})$ be an $r$ by $r$ matrix with entries in an arbitrary ring, and let $A^{*}=(a_{ij}^{*})$ be the adjoint matrix, i.e., $a_{ij}=(-1)^{i+j}det(A_{ij})$, ...
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1answer
21 views

When is the external tensor product of indecomposable modules indecomposable?

Let $k$ a field and $A$, $B$ finite dimensional associative $k$-algebras. If $M$ is an irreducible $A$-module, and $N$ is an irreducible $B$-module, then $M\otimes_kN$ is an irreducible $A\otimes ...
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1answer
15 views

Module over different rings

If $M$ is a finitely generated abelian group, which can be made into a module over a ring $R$ (with a certain scalar multiplication) and has as a module the minimal spanning set $\{e_1, \ldots, ...
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1answer
18 views

Using Exchange Lemma in a decomposition

The "Exchange Lemma" in the decomposition of modules states that any decomposition of right $R$-modules $M_1⊕...⊕M_n=A⊕B$ with the endomorphism ring $End (A_R)$ local yields $M_i≈A⊕X$ for some $i$, ...
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1answer
43 views

Exact sequence of modules and taking the quotient

Let $A$ be a commutative ring and $\text{Spec}\,A=\bigcup\limits_{i=1}^mD(f_i)$ be a covering by principal open sets. Show that the sequence of modules $$M\stackrel{\alpha}\to ...
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1answer
62 views

stationary set,club,module theory,Auslander lemma,

Here http://www.ams.org/journals/tran/1990-322-02/S0002-9947-1990-0974514-8/S0002-9947-1990-0974514-8.pdf I do not understand the first two lines of the proof of lemma 9,on page 550:What and why is ...
4
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1answer
79 views

The realationship of $\hom(M\otimes_RN,M'\otimes_RN')$ and $\hom_R(M,M')\otimes\hom_R(N,N')$.

Let $R$ be a ring with identity, $M$ and $M'$ two right $R$-module, $N$ and $N'$ two left $R$-module. There is a natural way to define a homomorphism ...
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2answers
32 views

Why this set is a $R/\mathfrak m$-module?

I'm reading this PDF: Discrete Valuation Rings and Function Fields of Curves. I'm trying to understand in this theorem why $\mathfrak m/\mathfrak m^2$ is a $R/\mathfrak m$-module (see number 4 below). ...
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0answers
34 views

The opposite of the right ideal in the ring of 2x2 matrices?

Since every ring has an opposite, I would like to know: Which is the opposite of the rings of $n \times n$ matrices? More specifically, of the $2 \times 2$ matrices. Is there an opposite for the ...
3
votes
2answers
88 views

For what kind of $R$-modules $M$ can we find an element $m\in M$ satisfing that $i:M\to M\otimes_R M, x\mapsto x\otimes m$ is an epimorphism?

Let $R$ be a commutative ring with identity and $M$ a $R$-module. I'm interested in under what condition we can find an element $m\in M$ satisfing that $i:M\to M\otimes_R M, x\mapsto x\otimes m$ is an ...
3
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2answers
87 views

Is there a relation between $End(M)$ and $M$ under tensor products?

Let $R$ be a commutative ring and $M$ be an $R$-module (If necessary, you can assume more conditions) Let $\phi$ be an $R$-endomorphism on $M$ and $\overline{\phi}:M^{\otimes n}\rightarrow M^{\otimes ...