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

learn more… | top users | synonyms

0
votes
1answer
25 views

How to determine $Hom (M,M)$ for an irreducible $R$-module $M$?

More exactly, I'm considering $R$ to be a finite dimensional $\mathbb C$-algebra. For any $R$-module $M$, the $\mathbb C$ vector space $Hom_R(M,M)$ contains scalar multiplication and hence contains ...
3
votes
0answers
41 views

Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
1
vote
1answer
31 views

Finding all $\mathbb{Z}$-submodules of $\mathbb{Z}$ with a complement

I am trying to find all the $\mathbb{Z}$-submodules of $\mathbb{Z}$ that have a complement Here with complement I mean: Def: Let $N \subset M$ be an $R$-Submodule. A submodule $N'$ is called ...
4
votes
3answers
88 views

What is the meaning of $x*\sqrt{3} \mod 1$?

What is the meaning of $x*\sqrt{3} \mod 1$? I'm trying to understand this: $$5( x*\sqrt{3} \mod 1) $$ If we talk about: $x=19,22,48,98$ what will be the result? I don't know how to calculate ...
1
vote
2answers
85 views

Finitely generated module over PID; Dummit and Foote, Exercise 12.1.12

Let $R$ be a PID and let $p$ be a prime in $R$. (a) Let $M$ be a finitely generated torsion $R$-module. Use the previous exercise to prove that $p^{k-1}M/p^kM \cong F^{n_k} $ where $F$ is the field ...
2
votes
1answer
12 views

homomorphism of product of modules

Given $(A_i)_{i\in I}$ and $B$ modules over a ring $R$, with $I$ infinite, $A_i\ne 0\;\forall i$ and $B\ne 0$, is it true that $$Hom_R(A_i,B)=0\;\forall i\implies Hom_R(\Pi_{i\in I} A_i, B)=0\quad?$$ ...
0
votes
1answer
15 views

prove that $HOM_R(\bigoplus _{s\in S}R_s ,M)$ and $\prod_{s\in S}M_s$ are isomorphic as $R$ modules.

Let $R$ be a ring, $M$ module over $R$ and $S$ a non empty set. Let $\mathrm{Hom}_R(M,N)$ be the group of $R$ module homomorphisms from $M$ to $N$. Denote $M_s:=M$ and $R_s:=R$. Prove that the ...
1
vote
1answer
102 views

Prove that if $\phi:M\rightarrow\prod_{i\in I} M/M_i$ is surjective then $M=M_j+\bigcap\limits_{i\neq j} M_i \: \forall j\in I$ [closed]

Let $M$ be an $A$-module, $\{M_i\}_{i\in I}$ a family of submodules of $M$ and let $\phi:M\rightarrow\prod_{i\in I} M/M_i$ defined by $\phi(x)=(x+M_i)_{i\in I}$. Prove: $\phi$ surjective ...
2
votes
1answer
70 views

When are two simple tensors $m' \otimes n'$ and $m \otimes n$ equal? (tensor product over modules)

Suppose that $M$ is a right R-module and $N$ is a left $R$-module. We can construct $M \underset{R}\otimes N$ and give it an Abelian group structure by considering the free R-module $K$ generated by ...
0
votes
1answer
58 views

References about “algebras over monoids”

Please, could someone point me any reference (with a bit of details) about "algebras over monoids" (in the sense of Schwede & Shipley, Algebras and modules in monoidal model categories)? Thank you ...
2
votes
2answers
49 views

Are projective modules “graded projective”?

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). ...
2
votes
1answer
41 views

Example of non-commutative ring with exactly 2014 two sided-proper ideals.

find a non-commutative ring with exactly 2014 two sided-proper ideals.find a ring with exactly 2014 pairwise non-isomorphic irreducible modules. If it was the commutative ring i would have thought of ...
0
votes
1answer
32 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 ...
1
vote
2answers
58 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 ...
2
votes
1answer
48 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. ...
1
vote
1answer
43 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 ...
0
votes
2answers
53 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 ...
1
vote
2answers
49 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 ...
1
vote
1answer
42 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 $A$ ...
1
vote
1answer
38 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 $\Bbb Z_2$ ...
1
vote
1answer
53 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 = ...
0
votes
1answer
21 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 ...
5
votes
1answer
68 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 ...
0
votes
0answers
24 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?
1
vote
1answer
19 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} ...
1
vote
2answers
28 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 ...
1
vote
1answer
34 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 ...
4
votes
1answer
54 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 ...
3
votes
1answer
58 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 ...
1
vote
1answer
33 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 ...
0
votes
2answers
16 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
0
votes
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) ...
0
votes
3answers
44 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 ...
1
vote
1answer
22 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
vote
1answer
39 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
21 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
75 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
82 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
48 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
50 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
31 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
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.
1
vote
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 ...
1
vote
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. ...
1
vote
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 $M$ ...
0
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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 ...