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

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1answer
24 views

Z_m module as Z_n module

I think every Z_n module can be viewed as Z_m module. I would like to confirm that we don't require any restriction on n and m, such as m divides n.
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1answer
34 views

Why an ideal in a ring is a submodule of a free module over that ring

I know an ideal in a ring is a module over this ring, but I don't know why it's a submodule of a free module, what's the free module? Thank you.
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1answer
65 views

Modules of Finite Length over Local Artinian Rings

Let $R$ be a commutative local artinian ring with identity. Denote its maximal ideal by $\mathfrak{m}$ and let $\mathbb{k}$ denote the residue field $\mathbb{k}=R/\mathfrak{m}$. Assume also that there ...
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0answers
47 views

Module and group ring: definitions and notations

I apologize in advance for the stupid questions and the bad English, but I've started studying math few months ago. I've some problems with the definitions of group ring, modules and their notations. ...
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1answer
37 views

Is a projective $R[G]$-module a projective $R[H]$-module if $H$ is a subgroup of $G$?

I have a ring $R$ of characteristic $0$ and a finite group $G$. Let $H$ be a subgroup of $G$. Question: If $M$ is a projective $R[G]$-module where $R[G]$ is the usual group ring then is $M$ ...
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0answers
25 views

Semisimple modules and short exact sequences

Though I suspect that this is a folklore result, I've been unable to find a proof by digital book searching. I also mention (my apologies) that I am new to noncommutative algebras theory and I am not ...
3
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1answer
67 views

Characterization of free modules

Let $M$ be a finitely generated module over a commutative ring $A$. Is it true that if there exists a positive integer $n$ and a pair of homomorphisms $\pi:A^n\rightarrow M$ and $\phi:A^n\rightarrow ...
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1answer
28 views

Finitely generated submodule of a localisation

Let $R$ be a commutative ring with unity, and $S$ be a multiplicatively closed set. Consider a $R$-module $M$ and a submodule $N\subset S^{-1}M$ (where $S^{-1}M$ is considered as a module over ...
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2answers
32 views

Property of unfaithful module over PID

Proposition Let $R$ be a PID and $M$ a torsion module over $R$. Suppose $M$ is not faithful, with $\operatorname{Ann(M)}=Ra$ and $a=u{p_1}^{\alpha_1}...{p_n}^{\alpha_n}$ an irreducible factorization ...
3
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1answer
100 views

Quick Question on a Proof of Artin-Wedderburn Theorem

Question [Edited]: [See below.] Are the isomorphisms in $(1)$ and $(2)$ (additive) group homomorphisms? If I'm right, $\text{End}_R(M)$ is a ring, but ...
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1answer
27 views

Trace ideal of generator

Let $R$ be a unital ring. In Lam's "Lectures on modules on rings" (Theorem 18.8, p483), the following implication is stated for a right $R$-module $P$: $$\mathrm{tr}(P) = R \implies R \text{ is a ...
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0answers
46 views

Syzygies of $Gr_{2}(\mathbb{C}^4)$

I'm going to start the study the problem of syzygies. I read on web that it is possible to compute syzygies of an algebraic variety for example $Gr_2(\mathbb{C}^4)$, but I don't understand how can I ...
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2answers
97 views

PID and finitely generated module

I am trying to prove the following statements: Let $R$ be a PID and $M$ a finitely generated $R$-module. Prove: (a) $M$ is torsion module iff $\operatorname{Hom}_R(M,R)=0$ (b) $M$ is an ...
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1answer
24 views

Show that there is an $R$-module homomorphism $\bar{h}$ such that $g \circ \bar{h} = h$.

Let $R$ be a unital ring. Suppose that a finitely generated free $R$-module is the internal direct sum of submodules $P$ and $Q$. Let $g:A \rightarrow B$ be a surjective $R$-module homomorphism. Let ...
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1answer
28 views

Some properties of matrices over rings

Problem (i) Let $R,T$ be division rings and $m,n \in \mathbb N$, then $M_n(R \times T) \cong M_n(R) \times M_n(T)$ and $M_m(M_n(R)) \cong M_{mn}(R)$. (ii) If $R$ is a semisimple ring and $n \in ...
2
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3answers
30 views

Torsion-free module morphism

I am trying to prove the statement: Let $R$ be a PID but not a field and let $M$ be an $R$-module. Then $$ M \space \text{is torsion-free $R$-module} \space \text{iff} \space ...
2
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1answer
58 views

PID and module problem

Let $R$ be a principal ideal domain but not a field, and let $M$ be an $R$-module. Show the following: (i) Let $p \in R$ be an irreducible element and $r \in R \setminus \{0\}$. Then $(R/ ...
2
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2answers
28 views

Extension of scalars $M_B=B\otimes_A M$

Most textbooks say that the $B$-module structure on $M_B$ (for $A\rightarrow B$ a ring morphism and $M$ an $A$-module) is "defined" by $b'(b\otimes m)=b'b\otimes m$. How is this a proper definition? ...
0
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1answer
55 views

A question on short exact sequences

Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N\to 0$ be a splitting short exact sequence so it exist a morphism $r: M \to L$ such that $r \circ \alpha = Id_L$ and a morphism $s: N \to M$ such ...
2
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1answer
61 views

Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
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1answer
71 views

Is there an interpretation of higher cohomology groups in terms of group extensions?

1) Consider a group $G$ and a $G$-module $A$. Then it is well-known that there is a $1-1$ correspondence between elements of $H^2(G,A),$ and group extensions $1\rightarrow A \rightarrow H\rightarrow ...
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0answers
30 views

Morphisms and Flatness

Do there exist morphisms $\phi \colon R \longrightarrow S$ of rings such that every $S$-module is flat as an $R$-module?
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2answers
65 views

Intersection of two flat submodules

Let $A$ be a ring, $M$ an $A$-module and $M_1,M_2$ two flat $A$-submodules of $M$. Is $M_1 \cap M_2$ a flat $A$-submodule of $M$?
3
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1answer
47 views

Characterize semisimple rings of quotients

Problem Let $K$ be a field. Characterize all polynomials $f\in K[X]$ such that $R=K[X]/\langle f\rangle$ is a semisimple ring. I know two equivalent definitions of semisimple modules but I am not so ...
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0answers
82 views

Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)

Q: (Exemplification) Do examples family of $R$-modules like $\{M_i\} , \{N_i\}$ and $R$-modules like $M,N$ such that every of four question in underline is true. (in other word for every question ...
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1answer
31 views

Do the $n\times n$ matrices over a division ring $D$ form a free $D$-module?

Let $D$ be a division ring. Then the set of $n\times n$ matrices over $D$ is free as a $D$-module. I think this is wrong because they are linearly dependent, right? But what does the ...
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0answers
20 views

Eisenstein Plane

Suppose I plotted points of the Eisenstein integers $\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$, where the point $a+b\omega$ was the point $(a,b)$. How could I describe this plane in the language ...
3
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1answer
62 views

Are all simple left modules over a simple left artinian ring isomorphic?

I've a basic question. $R$ is a simple left artinian ring. I want to show that all simple left $R$-modules are isomorphic. A simple $R$-module $M$ is isomorphic to $R/J$ where $J$ is a maximal ...
2
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1answer
54 views

Multiplicity of a dual simple module in the dual module?

Let $A$ be a finite dimensional $k$ algebra. Let $S$ be a simple left $A$-module and $M$ be any left $A$ module. Then my first question is that is it true $$[M:S]=[M^*:S^*]$$ where ...
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0answers
41 views

Is there any relationship between localization and completion of a module?

Let $R$ be a commutative ring, $\mathfrak p$ a prime ideal of $R$ and $M$ an $R$-module. I've seen the terms 'localization' $M_\mathfrak p$ of $M$ and the completion $M_\mathfrak p$ at $\mathfrak p$ ...
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0answers
37 views

Automorphism of the matrix algebra is an inner automorphism

How can I find an elementary proof for Skolem Noether's theorem which says that every algebra automorphism of the matrix algebra $M_n(K)$ is an inner automorphism. Thank you!
3
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1answer
70 views

Quotient of noncommutative algebra

Let $R=k\langle x,y,z \rangle$ be the non-commutative algebra in $3$ variables. Let $I$ be the ideal defined by the relations $xy-yx,yz-zy,xz-zx$. How to show formally that $R/I$ is the polynomial ...
1
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1answer
30 views

Semisimple submodule

Suppose that $M$ is a semisimple module, i.e $M \cong \oplus_{i \in I} S_{i}$ where $S_{i}$ are simple modules. Let $N$ be a submodule of $M$. Why there exists a subset $J \subseteq I$ such that $N ...
3
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1answer
71 views

Finding simple modules in fields

I am struggling with finding all simple are modules for general rings i know that the simple modules of R correspond with the simple modules of R/rad(R), where rad(R) is the Jacobson radical but i ...
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2answers
152 views

Product of Nilpotent ideal and simple module

I am stuck with trying to show that if an ideal I of a ring R is Nilpotent and M is a Simple R-Module, then IM = 0 I have attempted showing this by using the fact that the annihilator of a simple ...
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1answer
33 views

On the construction of an $R$-Module

Let $X \neq \emptyset$ be a set and $(R,+, \cdot)$ a commutative Ring with $\mathbb{1}$ and $(N,+, \cdot)$ an $R$-Module. Show that $(\text{map}(X,N), +, \cdot)$ is an $R$-Module where for $A= ...
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1answer
35 views

Prove $Rm$ simple $\iff Ann(m)$ is a left ideal of $R$.

I'm trying to that prove $Rm$ is a simple ring $\iff Ann(m)$ is a left ideal of $R$. I did a couple of questions earlier that I believe could be useful to show this: $(Q1)$ Quotient of cyclic ...
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1answer
27 views

Functorial isomorphism involving tensor products

Let $R$ be a commutative ring and $E', E, F', F$ be free, f.g. $R$-modules of equal rank. For $f\in L(E',E):={\rm Hom}_R(E',E)$ and $g\in L(F',F)$, let $T(f.g)\in L(E'\otimes_R F', E\otimes_R F)$ be ...
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0answers
23 views

Modules over PIDs

Let $M=\mathbb{Z}^4/N$ where $N$ is a subgroup of $\mathbb{Z}^4$ generated by $(1,0,-1,3)$ and $(2,4,8,-6)$. Recognize $M$ as a product of cyclic groups. Here I have to use the following Theorem: If ...
0
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1answer
31 views

Quotient of cyclic $R$-module is cyclic.

Let $R$ be a ring. I want to prove that every quotient of a cyclic $R$-module is cyclic. Can I assume that the question is about the quotient module or it could be just quotient? Because in the ...
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2answers
47 views

Prove isomorphism for a commutative ring

Show that $M\simeq \operatorname{Hom}_R(R,M)$ considering them as $R$-modules and $R\simeq \operatorname{Hom}_R(R,R)$ considering them as rings. I couldn't find a way to define the isomorphism.
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1answer
78 views

A simple module is necessarily the socle of its injective hull?

According to the wikipedia page on injective hulls "a simple module is necessarily the socle of its injective hull". It is clear to me that why a simple module is a subset to the socle of of its ...
1
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1answer
24 views

Intersection distributes over equality

Let $N$ be a submodule of an $R$-module $M$ and $I$ an ideal of $R$. Is the equality $$\frac{\bigcap_{n\ge 1}(I^nM+N)}{N}=\bigcap_{n\ge 1}(\frac{I^nM+N}{N})$$ true?
3
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1answer
65 views

Representatives of simple modules

For each of the following rings find a list of representatives of all simple $R$-modules: 1) $R=\mathbb C[x]$ 2) $R=\mathbb R[x]$ What I've tried was: I know that $M$ is a simple ...
2
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1answer
60 views

Let $R$ be a $M\times N$ matrix with rational entries. Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under $R$. Consider an equivalence relation on $R\mathbb{Z}^N$ defined by $a\sim b$ if $a-b\in ...
2
votes
1answer
21 views

Boundary Homomorphism

I was studying the proposition 2.10 of Atiyah and MacDonald's Introduction to Commutative Algebra, and have a question. The proposition says: Let $$ \require{AMScd} \begin{CD} 0 @>>> ...
2
votes
3answers
117 views

Is every submodule of a projective module projective?

Is every submodule of a projective module projective? I know that the answer is no, but I haven't been able to come up with any concrete examples despite quite a bit of effort. Also, if the ...
3
votes
4answers
88 views

If every free $R$-module has the property that independence implies extendibility, is $R$ necessarily a field?

Definition. Whenever $M$ is a free $R$-module, let us call a subset $A$ of $M$ extendible iff there is a basis $B$ for $M$ such that $A \subseteq B$. (Is there a standard name for this condition?) ...
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0answers
44 views

How to compute $Ext_A^{1}(S_1, S_2)$ and $Ext_A^{1}(S_2, S_1)$?

Let $A = kQ/\rho $, $Q$ is the quiver \begin{align} 1 \overset{a}{\underset{a^*}{\rightleftarrows}} 2 \end{align} $\rho$ is the relation $a a^* - a^* a = 0$. Question: compute $Ext_A^{1}(S_1, S_2)$. ...
2
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1answer
47 views

Mitchell's Embedding Theorem for not-necessarily-small categories

Mitchell's Embedding Theorem states that if $\mathcal{A}$ is a small abelian category, then there is a ring $R$ and a fully-faithful exact functor $F:\mathcal{A}\rightarrow R\mathsf{Mod}$. To what ...