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

learn more… | top users | synonyms

1
vote
1answer
21 views

When is the Tensor product of Modules itself a Module?

If $M$ is a right $R$ module and $N$ is a left $R$ module then $M \otimes_R N$ is an abelian group. Wikipedia says that if $M$ is an $R$ bimodule then $M \otimes_R N$ can take on the structure of a ...
-1
votes
0answers
37 views

Show that field of fraction of a commutative domain is an indecomposable module which is not finitely generated

I came across this problem and get stuck for quite sometime. Problem: Let $R$ be a commutative domain that is not a field. Let $F$ be its field of fractions. Show that $F$ is an indecomposable ...
1
vote
1answer
38 views

Showing the socle of a module exists and is unique

I have been set the following exercise: For every $R$-module $M$, show that there exists a unique semi-simple submodule $sM$ $\subset$ $M$ which contains every semi-simple submodule of $M$. ...
0
votes
0answers
15 views

Ring of Upper triangular matrices submodule

Suppose that $R$ is the ring of all matrices of the form $ \left( \begin{array}{cc} z & p \\ 0 & q \end{array} \right) $, where $z\in \mathbb{Z}$ and $p,q\in \mathbb{Q}$. Let $$ I_n= ...
3
votes
2answers
65 views

Direct summand of modules proof

I'm have problems trying to show this, any help would be appreciated. Show $i: N \subset M$ is a direct summand iff $\exists$ a module map $r: M \to N$ s.t $ri = 1_{N}$, and that any complement of ...
0
votes
1answer
68 views

Is $R$ PID if every submodule of a free $R$-module is free?

Let $R$ be a commutative ring. Before I proved that every submodule of a free $R$-module is free over a P.I.D. Now I'm trying to prove the reciprocal, if every submodule of a free $R$-module is ...
1
vote
1answer
47 views

Examples of modules (basis and generators)

I have to give examples for modules such that $1.$ There's a linearly independent set with more elements than a generator set. $2.$ There's a linearly independent set with the same cardinal than a ...
0
votes
0answers
47 views

Cardinal of a linearly independent subset of $R$-module

Let $R$ be a commutative ring, and consider $R$ as an $R$-module with the action given by the product of $R$. Prove that if $B\subset R$ is linearly independent, then $\operatorname{card}(B)=1.$ ...
1
vote
1answer
29 views

Identifying sets

I keep seeing written in texts the phrase 'identify sets', for example: Identify $A$ as a subset of $F(A)$ by $a\mapsto f_a$, where $f_a$ is the function which is 1 at a and 0 elsewhere. This is in ...
0
votes
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.
0
votes
1answer
30 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.
1
vote
1answer
59 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 ...
1
vote
0answers
43 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. ...
1
vote
1answer
35 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 $M$ ...
0
votes
0answers
21 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
votes
1answer
66 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 ...
1
vote
1answer
67 views

Length of a module over a local Artinian ring

If $R$ is a local Artinian ring with maximal ideal $P$, and $M$ is a finitely generated $R$-module, I would like to show that the length of any series $M= M_n\geq M_{n-1}\geq\dots\geq M_0=(0)$ such ...
1
vote
1answer
23 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 ...
0
votes
2answers
29 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
votes
1answer
94 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 ...
2
votes
1answer
25 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 ...
2
votes
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 ...
3
votes
2answers
93 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 ...
0
votes
1answer
23 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 ...
0
votes
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
votes
3answers
23 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
votes
1answer
53 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
votes
2answers
25 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
votes
1answer
53 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
votes
1answer
54 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 ...
4
votes
1answer
68 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 ...
0
votes
0answers
29 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?
0
votes
2answers
61 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
votes
1answer
46 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 ...
2
votes
0answers
74 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 ...
1
vote
1answer
29 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 ...
1
vote
0answers
17 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 ...
2
votes
1answer
46 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
votes
1answer
52 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 ...
1
vote
0answers
40 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 $\mathfrak p$ ...
1
vote
0answers
26 views

Direct limit of injective modules are injective for left Noetherian rings

This is an exercise in Cartan Eilenberg's Homological algebra. Let $\Lambda$ be a left Noetherian rings, then the direct limit of left injective modules is injective. My solution: Let $I$ be ...
0
votes
0answers
34 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!
1
vote
1answer
60 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
vote
1answer
27 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
votes
1answer
70 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 ...
6
votes
2answers
148 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 ...
0
votes
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= ...
1
vote
1answer
31 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 ...
0
votes
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 ...
1
vote
0answers
22 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 ...