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

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3
votes
1answer
21 views

Is an injective map of $B$-modules also injective as an $A$-linear map if $B$ is an $A$-algebra?

I've been going through my submitted exercises again of my Commutative Algebra-class and I have the following question: Let $A$ be a commutative ring with unity. Given any injective homomorphism of ...
1
vote
0answers
27 views

Constructing a ring consisting of formal infinite series from a given ring

Let $A$ be an $\mathbb{N}$-graded $\Bbbk$-algebra, where $\Bbbk$ is a field, and where $\dim_\Bbbk A_n < \infty$ for all $n \in \mathbb{N}$. I can't see anything preventing me from constructing a ...
0
votes
2answers
49 views

Prove that $\operatorname{Ext}^{d+1}(A, B)\cong \operatorname{Ext}^1(M_d,B)$

So, given a resolution, with $P_{i}$ projective modules: $$0\longrightarrow M_d\longrightarrow P_{d-1} \longrightarrow \cdots \longrightarrow P_0 \longrightarrow A\longrightarrow 0,$$ I'm trying to ...
7
votes
2answers
131 views

Problem with the ring $R=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q &\Bbb Q\end{bmatrix}$ and its ideal $D=\begin{bmatrix}0&0\\ \Bbb Q & \Bbb Q\end{bmatrix}$

Let us consider the ring $ R:=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q & \Bbb Q\end{bmatrix} $ and its two-sided ideal $ D:=\begin{bmatrix}0 & 0\\ \Bbb Q & \Bbb Q\end{bmatrix} $. Let then ...
0
votes
2answers
24 views

Quotient space decomposition

Let $V$ be a vector space(or module) with decomposition $V= V_1 \oplus V_2$. And let $W \subset V$ be a subspace with decomposition $W= W_1 \oplus W_2$ such that $W_1 \subset V_1$ and $W_2 \subset ...
0
votes
2answers
37 views

Checking commutativity of a diagram of modules over some ring and what the commutativity of the diagram implies.

Suppose that you have the following diagram of modules over some ring: These are my questions: (1) To prove that the diagram is commutative, we needs to prove that $gf=kh$, $wf=rv$, $zh=uv$, ...
3
votes
1answer
50 views

Category of modules

For given a ring $R$, we can define the category of left $R$-modules. In fact, objects are all left $R$-modules and morphisms are $R$-module homomorphisms. NOw my question is: If we do not fix $R$, ...
2
votes
1answer
32 views

Kernel of a natural map is a direct summand of the covariant extension

I am reading chapter 2 of 'Homological Algebra' by Cartan and Eilenberg. 1/ Given a ring homomorphism $\varphi: \Lambda \rightarrow \Gamma$ and a right $\Gamma$-module $A$, we can treat $A$ and ...
1
vote
0answers
43 views

Free $R$-module when $R$ is not unital

We can easily construct free $R$-module when $R$ is unital by setting $$R[S] = \{ f\colon S\to R\,|\, f\ \text{finitely supported}\}$$ and defining operations pointwise. The key here is that we can ...
1
vote
2answers
65 views

Definition of exact sequence [closed]

Take a sequence $A \to B \to C \to 0$ of modules over a commutative ring. How would one show that it is exact? I understand the necessary surjectivity of $B \to C$, but what about the first map?
0
votes
0answers
47 views

Comodule induced from a comodule over a graded coalgebra

Let $G$ be a group with $1$ be the identity element of $G$. Let $(C,\bigtriangleup,\epsilon)$ be a coalgebra over a field $K$. $C$ is called $G$-graded if $C$ admits a decomposition as a direct sum of ...
3
votes
2answers
51 views

Example of a $\mathbb{Z}$-module with exactly three proper submodules?

What is an example of a $\mathbb{Z}$-module which has exactly three proper submodules?
1
vote
1answer
40 views

$\mathcal{O}_L$ free over $\mathcal{O}_K[G]$

Let $L/K$ be a galois extension of number fields. Suppose $G:=\text{Gal}(L/K)$ is abelian. If $\mathcal{O}_L$ is free as $\mathcal{O}_K[G]$-module, is it true that it has rank 1?
3
votes
3answers
69 views

Equivalent definition of R-module

I am studying a course of Commutative Algebra and I'm having trouble understanding an alternative definition of R-module. The standard definition is the following: Let $R$ be a ring (commutative, ...
0
votes
1answer
83 views

How do I prove (Module isomorphic direct sum of modules)

Module $M$ isomorphic to the external direct sum of modules $\bigoplus_{\alpha\in I}M_{\alpha}$ if and only if there are monomorphisms $i_{\alpha} : M_{\alpha} \rightarrow M, \alpha \in I$, such that ...
0
votes
0answers
28 views

What is the use of a right-module?

It seems that only the left-module provides a representation of a group. So what is the use of a right-module?
3
votes
0answers
40 views

Proving that Tor is a balanced functor using the derived category

At this end of this expository article on derived categories, R.P. Thomas says the following. There are two main advantages of this approach. Firstly that we have managed to make the complex ...
2
votes
1answer
56 views

What is a periodic module

I've been reading the text "THE THEORY OF COMMUTATIVE FORMAL GROUPS OVER FIELDS OF FINITE CHARACTERISTIC" by Manin. On page 26, proposition 2.1, the author mentions the notion of a periodic module, ...
3
votes
1answer
28 views

How do you break up an exact sequence of any length to a “succession of short exact sequences”?

Note that if $\text{Hom}_R(D,-)$ functor takes short exact sequences to short exact sequences then it takes exact sequences of any length to exact sequences since any exact sequence can be broken ...
1
vote
3answers
48 views

Doubts about adjunction of elements to a ring

I've just solved the following exercise Suppose we adjoin an element $\alpha$ to $\mathbb{R}$ satisfying the relation $\alpha^2 = 1$. Prove that the resulting ring is isomorphic to the product ...
-1
votes
1answer
28 views

Localization $U^{-1} N$ where $U = R^{\times}$ is the set of nonzero elements of an integral domain.

Suppose $R$ is an integral domain with quotient field $Q$ and let $N$ be any $R$-module. Let $U = R^{\times}$ be the set of nonzero elements in $R$ and define $U^{-1}N$ to be the set of ...
-2
votes
1answer
57 views

Can we have $N\oplus P=N' \oplus P$ for $N' \subsetneq N$?

Let $M$ be a $R$-module. Is it possible that we have two submodules $N,P \subset M$ such that $M=N\oplus P$ and also $M=N'\oplus P$ for some $N' \subsetneq N$? If $R$ is a field and $M$ a finite ...
-2
votes
1answer
24 views

Uniform modules and submodules [closed]

A non-zero left module $M$ over a ring with unity is called uniform if any two non-zero submodules have non-empty intersection; $M$ is said to contain enough uniforms if any non-zero submodule ...
0
votes
1answer
36 views

Simple submodule of modules

I know that a simple right module is a non-zero right module $M_R$ whose submodules are only $M_R$ and $0$. Now fix a module $N_R$. If I want to show that a submodule $M_R$ of $N_R$ is simple what do ...
1
vote
0answers
22 views

Structure theorem for divisible modules

I would like to know if there is some analogue theorem of structure for divisible modules as there is for divisible abelian groups. More exactly, given a divisible $R-$ module M, where $R$ is a PID, ...
2
votes
1answer
36 views

Can a graded $k$-algebra have torsion over $k[\theta]$ for $\theta$ a non-zerodivisor?

Let $k$ be a field and let $R$ be an $\mathbb{N}$-graded $k$-algebra such that $R_0=k$. Let $\theta$ be a not necessarily homogeneous element of $R$ that is transcendental over $k$ and is a ...
2
votes
3answers
27 views

Equality of (sub)modules with a specific exact sequence of modules

When reading these notes of Andy McLennan I came across Lemma A4.3 and it's proof. $\textbf{Lemma 4.3}.$ If $0\to L\xrightarrow{f}M\xrightarrow{g} N\to 0$ is a short exact sequence of $R$-modules, ...
2
votes
2answers
65 views

$\mathrm{Hom}_R(N,M)$ is an essential extension of $\mathrm{Hom}_R(N,L)$

$M$ is an $R$-module and an essential extension of $L$. $N$ is a finitely generated submodule of $M$. Then $\mathrm{Hom}_R(N,M)$ is an essential extension of $\mathrm{Hom}_R(N,L)$. I tried to use ...
1
vote
0answers
41 views

Slick proof of isomorphism of quotients of direct sums of modules

I have just encountered this problem on quotients of direct sums of modules and their isomorphisms stating as follows: For all that follows R is a ring with unity (not necessarily commutative) and ...
0
votes
0answers
33 views

Characters of modules of associative algebras

Let $F$ be a field, and let $A$ be a finite-dimensional associative $F$-algebra with multiplicative identity 1. If $M$ is a finite-dimensional module of $A$, define the character $\chi_M:A\rightarrow ...
1
vote
0answers
36 views

Alternative isomorphism map to prove R-module isomorphism

To prove $(R/I)\otimes_R M\cong M/IM$ ($R$ a commutative ring with 1, $I$ an ideal in $R$, $M$ is an $R$-module.) My professor's solution involves a map $f:(R/I)\otimes_R M\to M/IM$, with ...
2
votes
2answers
42 views

Isomorphism of free modules relating to matrices

I have recently encountered this question in my Abstract Algebra studies, which states: Let $R$ be a ring with unity and for two natural numbers $ m,n \in\mathbb N $ for which the two free ...
2
votes
2answers
48 views

Tensor product of two $R$-modules

$R$ a ring. Given $M$ a right $R$-module and N a left $R$-module, we define the tensor product $$M \otimes_R N$$ where (among other properties) $mr \otimes n = m\otimes rn$ holds for all $r \in R, m ...
4
votes
1answer
47 views

In what generality does every module show up over some residue field?

Let $R$ be a commutative, unital ring and $M$ an $R$-module. It is well-known that if $M$ is nonzero, there is some prime $\mathfrak{p}\in \operatorname{Spec} R$ with $M\otimes_R R_\mathfrak{p}$ ...
0
votes
0answers
30 views

Dual of a comodule is a comodule in case that is the module is finitely generated and projective

Let $R$ be a commutative ring with identity. Let $(C,\bigtriangleup,\epsilon)$ be an $R$-coalgebra where $\bigtriangleup$ is the coproduct and $\epsilon$ is the counit. Assume that $(P,e^{p})$ be a ...
0
votes
1answer
38 views

Finite separable extension of a field, free module.

I'm trying to understand proof of following theorem. $\textbf{Theorem.}$ If $L$ is finite and separable extension of field $K$, $K$ is field of fractions of principal ideal domain $A$ and $B$ is ...
2
votes
1answer
44 views

Finite dimensional algebras with finite global dimension.

Let $A$ be a finite dimensional $k$-algebra, $k$ is a field, with a finite global dimension. I wonder if that implies $A$ is tame or finite type? or more generally is there a relation between these ...
4
votes
1answer
33 views

Show that $Hom_R(R^n,M) \cong M^n$ for R-modules

We want to show that $Hom_R(R^n,M) \cong M^n$ for $n\in\Bbb Z_{\ge0}$ I have already shown that $Hom_R(R,M) \cong M$ by letting $f:Hom_R(R,M)\rightarrow M$ given by $f(\phi) = \phi(1)$. I showed ...
4
votes
1answer
31 views

If $F\cong R^n$, why is $1\otimes \psi\colon F\otimes_R L\to F\otimes_R M$ injective?

Suppose that $\psi\colon L\to M$ is an injective homomorphism of $R$-modules and $F\cong R^n$. I would like to show that $1\otimes \psi\colon F\otimes_R L\to F\otimes_R M$ is also injective (this is a ...
1
vote
1answer
29 views

Relating generating sets of modules over quotient ring to modules over original ring

I am having trouble with the following exercise. Let $R = F[x]$, where F is some field. Let $I = (x^2)$, and let M be an R/I-module (induced action, so assuming $IM = 0$). Then I want to show that ...
6
votes
1answer
84 views

Module which is not direct sum of indecomposable submodules

I would like to find an example of a ring $R$ and a $R$-module $M$, which can't be written as a direct sum of indecomposable submodules, i.e. $$ M \not \cong \bigoplus\limits_{i \in I} M_i$$ ...
0
votes
2answers
28 views

Must an artinian R-module have a simple R-submodule?

Let $M$ be an artinian $R$-module, then by the definition; for every descending chain of $R$-submodules $M_i, \: (i = 0,1,2,...)$ of $M$, the chain must become stationary for some $i=k$. Now I assume ...
1
vote
1answer
26 views

Let $R = M_n(k)$, where $k$ is a field. Then any R-module that is finite dimensional over K is a direct sum of

Let $R = M_n(k)$, where $k$ is a field. Then any $R$-module that is finite dimensional over $K$ is a direct sum of isomorphic copies of $V$, where $V = k^n$. I was able to show that $R$ has a ...
0
votes
1answer
46 views

Saturated submodule

I'm trying to solve this exercise, but I can't... Let S be a multiplicatively closed subset of A, M an A-module, N a submodule of M and $\phi_M:M\longrightarrow S^{-1}M$ the homomorphism given by ...
1
vote
0answers
48 views

Indecomposable, infinitely generated, non-isomorphic modules over the integers

I am having trouble proving the following statement. Let $A$ and $B$ be infinite, disjoint sets of prime numbers with $5 \notin A\cup B$. Set $e_2' = 5e_1+2e_2$ where $e_1,e_2,e_3$ denotes the ...
0
votes
1answer
76 views

$\mathrm{Ann}(I \cap J) = \mathrm{Ann}(I) +\mathrm{Ann}(J)$ if $R$ is self-injective

If $R$ is injective as an $R$-module, then for every two ideals $I$ and $J$ we will have $\mathrm{Ann}(I \cap J) = \mathrm{Ann}(I) + \mathrm{Ann}(J)$. I made an effort for an hour but it was not ...
1
vote
1answer
14 views

On indecomposable modules example

Am preparing for exam few days to go. I came across this problem in Anderson - Fuller book about modules. (1) Give an example of an indecomposable module that has a decomposable submodule. (2) Give ...
1
vote
2answers
59 views

Proof of Proposition 2.12 in Neukirch ANT

I'd like a reference or a direct proof of the following statement: Let $K|\mathbb Q$ be a finite extension and consider the ring of algebraic integers $\mathcal O_K$. Let $\mathfrak ...
1
vote
1answer
42 views

Show that character vanishes on specific element $g$ if for $H \le G$ we have $[\chi_H, 1_H] = 0$ and all elements of $Hg$ are conjugate in $G$

Let $G$ be a finite group and $H \le G$ with $g \in G$ such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb C$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show ...
2
votes
2answers
40 views

Does an algebra over a ring with unity have unity?

Definition from ProofWiki: An algebra over a ring $(G_R,\oplus)$ is an $R$-module $G_R$ over a commutative ring $R$ with a bilinear mapping $\oplus:G^2→G$. Context: I want to show that an algebra ...