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

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11 views

Difference between to Tensor products with regards to modules

What would be the difference between $$ \otimes_B $$ and $$ \otimes $$ both in the following context and in general? Let A be a ring with $$ B \subset A $$ and M a B-Module. We can construct the ...
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3answers
27 views

Show that $\operatorname{Hom}(S(-d),S)\cong S(d)$ where $S$ is polynomial ring?

As stated above, $S$ is polynomial ring, and since the polynomial ring is $S$ and $S(-d)$ are finite over $S$ as graded modules, we can say that $\operatorname{Hom}(S(-d),S)$ is also graded. My ...
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1answer
29 views

Direct sum of simple modules and Schur's Lemma

Suppose $M,N$ are two non-isomorphic simple $R$-modules. For $m,n\geq1$, is it true that $$ \text{Hom}_R(M^{\oplus m},N^{\oplus n})\cong\hat{0}\,? $$ I think it's true by Schur's Lemma. ...
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13 views

Left R-module structure

Let $X$ be any non-empty set and let $S$ be a ring with $1$. Let $T$ denote the $S$-bimodule generated by $X$. Now let $R$ be another ring with $1$ and suppose $X$ is a left $R$-module. Can we view ...
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1answer
35 views

A finite-dimensional $K$-algebra?

Let $A$ be a commutative noetherian domain of characteristic $p>0$ and $K$ be its quotient field. Let $G$ be any finite group whose order is divisible by $p$. Is $KG$ a finite-dimensional ...
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2answers
43 views

Looking for help, Aluffi Exercise 5.13, Chapter 6: characterization of PIDs

I quote: "Let $M$ be a finitely generated module over an integral domain $R$. Prove that if $R$ is a PID, then $M$ is torsion-free if and only if it is free. Prove that this property characterizes ...
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2answers
36 views

Trouble on Aluffi's Exercise 5.5, Chapter 6: finitely generated projective module over a local ring is free

The overall goal of the problem is the following: Let $R$ be a local commutative ring (so that there is a unique maximal ideal $\mathfrak{m}$) and let $M$ be a finitely generated $R$-module. Assume ...
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1answer
34 views

What is ker $f \otimes g$?

I am trying to find out the $ker(f\otimes g)$ where $f:M \rightarrow P$ and $g:N \rightarrow Q$ are $A$ linear maps where $A$ is not a field.So $(f\otimes g):M\otimes N\rightarrow P\otimes Q $ is $A$ ...
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1answer
32 views

Isomorphism with tensor product

Let $f$ : $\Bbb Z_2$ $\rightarrow$ $\Bbb Z_4$ given as $f$($a$ + $2$$\Bbb Z$) = $2a$ + $\Bbb Z_4$ is a monomorphism . And knowing that <0,2> as subgroup of $\Bbb Z_4$ is isomorphic to $\Bbb Z_2$ ...
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1answer
36 views

Is a vector of coprime ring elements column of an invertible matrix?

Given a commutative ring $R$ with unit and $a_1=(r_1,\ldots,r_n)^T \in R^n$ with coprime entries (i.e. $\sum_i Rr_i=R)$. Are there $a_2,\ldots,a_n \in R^n$ such that the matrix $A = ...
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1answer
16 views

Rational Canonical Form Confusion; Choosing Basis Which Gives the Rational Canonical Form.

I am reading the theory of finitely generated modules over a PID. One of the applications of the the theory is that one can derive the theory of rational canonical form of a linear operator on a ...
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1answer
60 views

Why $R^{op}$ used in the definition of “category of $R$-modules”?

Earlier today, I was thinking: "Oh, an $R$-module is just an additive functor $R \rightarrow \mathbf{Ab}.$" Anyway, I had a bit of a read over at nLab, and it says: For any small ...
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0answers
20 views

Center of a ring projective?

If $R$ is a ring and $Z(R)$ denotes $R$'s centre, then when is $R$ projective as an $Z(R)$-module?
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1answer
13 views

Proof that the connecting morphism in the snake lemma is well defined

The snake lemma says: suppose we have two exact sequences of $R$-modules $M_1 \xrightarrow{f_M} M_2 \xrightarrow{g_M} M_3 \rightarrow 0$ $0\rightarrow N_1 \xrightarrow{f_N} N_2 \xrightarrow{g_N} ...
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2answers
26 views

If $M$ is a $R$-module with $R=R_1\times \cdots \times R_n$ then $M\simeq M_1\times \cdots \times M_n$ where $M_i$ is a $R_i$-module?

Let $R_1, \ldots, R_n$ be rings and consider the ring $R:=R_1\times \cdots\times R_n$. How can I show every $R$-module $M$ is isomorphic to a product $M_1\times \cdots\times M_n$ where each $M_i$ is a ...
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24 views

Why most discussion on ring(like module) is over PID, not UFD? [closed]

I think many properties and discussion are based on unique factorization. Like On UFD, irreducible element is also prime. So why these analysises are not based on UFD?
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1answer
32 views

A group ring has finite length

I have been given a version of Maschke's Theorem to prove: Let $k$ be a field and $G$ a finite group s.t. $|G|$ is non-zero in $k$. Show that the group ring $k[G]$ is a semi-simple ring. A hint ...
3
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1answer
42 views

Socle of submodule relative to the module

in these notes i am reading i am told that the socle of $K$ (where $K \subset M$ , and $M$ is a module) is = $K \cap$ Soc $ M$ But why is this? i see the intuition but cannot formalize a proof ...
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1answer
46 views

Homomorphisms of semi-simple submodules

I managed to show $\forall$ R-Module M, $\exists$ unique semi-simple submodule $sM \subset M$ containing every semi-simple submodule of M, by showing that the direct sum of semi-simple submodules are ...
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23 views

$I^{n-1}/I^n$ is a Noetherian $R$-module

Let $R$ be a commutative ring with unity and $I \subseteq R$ an ideal. Prove: if $R/I$ is a Noetherian ring and $I/I^2$ is a finitely-generated $R$-module, then $I^{n-1}/I^n$ is a Noetherian ...
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2answers
12 views

All the solutions for this system 5x+33y = 6 (mod 13) and 7x + 2y = 9 (mod 13)

I want all the solutions for this system. 5x + 3y = 6 (mod 13) and 7x + 2y = 9 (mod 13)... Thanks
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1answer
19 views

$f:M\to N$ is surjective $R$-linear map

$R$ be a commutative ring with $1$ and maximal ideal $m$.Let $f:M \rightarrow N$ be a $R-linear $ map and $N$ is finitely generated $R-$ module.Suppose that the induced homomorphism $M\otimes_R(R/m) ...
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3answers
33 views

When considering the commutative ring $R$ over itself, if its submodule is free, then the submodule equals to the module?

When considering the commutative ring $R$ over itself, then this $R$-module is isomorphic to $R$, but if $I$ is an ideal of $R$, then it is a submodule, if this submodule is free too, then it is ...
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1answer
20 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 ...
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1answer
35 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 ...
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1answer
36 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 $M$. ...
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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= ...
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2answers
60 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 ...
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1answer
57 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 ...
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1answer
46 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 ...
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39 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 $\operatorname{card}(B)=1.$ ...
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1answer
26 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 ...
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1answer
23 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
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.
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1answer
52 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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42 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
29 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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20 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 ...
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1answer
57 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
57 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 ...
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1answer
22 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
22 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 ...
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50 views
+100

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
22 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
45 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
87 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
18 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 ...
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3answers
21 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 ...
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1answer
28 views

A ,B is finitely generated R-algebra then $A\otimes_RB$ is fg $R$-algebra.

$A$,$B$ are finitely generated $R$-algebra.$R$ is a commutative ring with $1$.Then how can I show that $A\otimes_RB$ is finitely generated $R$-algebra. What I have tried: First I have to show that ...