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

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3
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1answer
175 views

Associated prime ideals of Hom (Bruns and Herzog, exercise 1.2.27)

Let $R$ be a Noetherian ring and $M,N$ finitely generated modules. I want to show that $$\mathrm{Ass}_R(\mathrm{Hom}_R(M,N)) = \mathrm{Ass}_R(N) \cap \mathrm{Supp}(M).$$ I don't understand what ...
3
votes
1answer
43 views

Question about proof of Corollary 2.18 from Eisenbud

I am reading Eisenbud's Commutative Algebra. The following is the proof I am trying to understand. My question is the second sentence in the proof. I understand that a power of $P_P$ annihilates ...
6
votes
1answer
47 views

Need for projective modules

I wanted to ask why we require projective modules. After studying all the essential ingredients my guess is - Firstly, we worked with vector spaces (say modules over field $F$) (which are free ...
2
votes
0answers
27 views

How to write isomorphisms $A^* \otimes B \cong \hom(A, B)$ and $\hom(A\otimes B, C) \cong \hom(A, B^* \otimes C)$ explicitly?

I would like to write isomorphisms $A^* \otimes B \cong \hom(A, B)$ and $\hom(A\otimes B, C) \cong \hom(A, B^* \otimes C)$ explicitly. I think that $\varphi: A^* \otimes B \to \hom(A, B)$ is given by ...
2
votes
2answers
35 views

Conceptual doubt about Modules

Let $R$ be a ring. Then a left $R$- module is an additive Abelian group M along with an operation $R\times M \to M$ such that it satisfies - $r(x+y)=rx+ry$ $(r+s)x=rx+sx$ $(rs)x=r(sx)$ $1_R x=x$ ...
3
votes
1answer
251 views

A example of a monoidal non symmetric category of $R$-bimodules

It is well know that if $R$ is a commutative ring, then the category of modules $_R\operatorname{Mod}$ is monoidal symmetric. (Edit. I wrote the category of $R$-bimodules, as F. Muro explained this ...
3
votes
1answer
171 views

There are no maximal $\mathbb{Z}$-submodules in $\mathbb{Q}$

Is it true that $\mathbb{Q}$ viewed as $\mathbb{Z}$-module (i.e. abelian group) has no maximal $\mathbb{Z}$-submodules? Why ?
0
votes
1answer
34 views

Zorn's Lemma Application for Finding Maximal Submodule?

I came across the following exercise in my algebra textbook. Let $M$ be a left $R$-module ($R$ is ring with unity $1_R$) and let $N\subseteq M$ be a submodule such that $x\in M-N$ for some $x\in ...
3
votes
2answers
29 views

Action of the endomorphism ring of a module

Let $A$ be a ring with $1$ and $M$ a left $A$-module. In chapter 6 of Curtis and Reiner's 'Methods of Representation theory', $M$ is regarded as a right module over the endomorphism ring ...
2
votes
1answer
37 views

Is $K[[x]]$ an Artinian/Noetherian $K[x]$-module?

Let $K$ be a field an consider $K[[x]]$ as a $K[x]$-module. Determine if it is Artinian/Noetherian. I used the following propositions: If M is an $R$-module and $N\subseteq M$ a submodule, then ...
7
votes
2answers
168 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
16
votes
4answers
1k views

Reference request: compact objects in R-Mod are precisely the finitely-presented modules?

Let $R$ be a ring. According to this MO question, the modules $M \in R\text{-Mod}$ such that $\text{Hom}(M, -)$ preserves all filtered colimits (the compact objects) are precisely the ...
2
votes
1answer
20 views

If $ R = L \oplus N $, then $ L = Re$ for idempotent $ e \in R $

I'm trying to prove the following statement. Suppose $ L \lhd R$ is a left ideal in $ R $ (a ring with unity). If there exists a left ideal $ N \lhd R $ such that $ R = L \oplus N $, then $ L = Re $ ...
0
votes
1answer
30 views

Simple problem about morphism in abelian categories

$f$ : $X\to$ $Y$ and $g$ : $Y\to$$Z$ a sequence in abelian categories. Show that if $gf$=$0$ if and only if exist a monomorphism $h$:$Im(f)$ $\to$ $Ker(g)$ such $kh$=$j$, where $j$:$Im(f)$$\to$ $Y$ ...
1
vote
1answer
34 views

A question on modules of rings

Before arising my question, let us see the following fact. Let $R$ be a ring with unity and $R'$ be a subring of $R$. If $M'$ is a simple $R'$-module, there exists a simple $R$-module $M$ such that ...
2
votes
2answers
39 views

Simple K[G] module has dimension 1 over K

Let G be a finite abelian group, K an algebraically closed field, K[G] the group algebra of G over K and M a K[G] module. I would like to show that if M is simple, it has dimension 1 over K. I think ...
4
votes
2answers
173 views

What exactly is a trivial module?

Yes, this is a quite basic answer, but I have to admit to be absolutely confused about this notion. Searching on the web, I managed to found two possible definition of trivial modules, referring ...
1
vote
1answer
86 views

Homotopy equivalence and chain complexes

This is from the book: Hilton and Stammbach, A Course in Homological Algebra, Chapter IV, Derived Functors, exercise 4.2. Let $\varphi:C \to D$ be a chain map of the projective complex $C$ into ...
1
vote
0answers
33 views

Relation between Extensions and self-Extensions!

This should be considered as very general question regarding the extension group $Ext^i _A (R,S)$, in particular where $i=1$, for $R$ and $S$, a pair of given objects in an abelian category $A$. For ...
0
votes
0answers
13 views

Indecomposables generate the algebra

The following result seems intuitive, but I'm having hard time proving it, or determining if it is, in fact, false. Let $(A, \mu, \eta, \epsilon)$ be an augmented algebra, with augmentation ideal ...
1
vote
2answers
55 views

Modules isomorphism

Studying vector spaces, we can find the well known result that every vector space of dimension $n$ over a field $k$ is isomorphic to $k^n$. Is there a similar theorem for modules? Thanks guys!
0
votes
0answers
33 views

Row-Column Expansion

I saw a Proposition as follows: Let $A=(a_{ij})$ be an $r$ by $r$ matrix with entries in an arbitrary ring, and let $A^{*}=(a_{ij}^{*})$ be the adjoint matrix, i.e., $a_{ij}=(-1)^{i+j}det(A_{ij})$, ...
2
votes
1answer
38 views

Reference Request: Replacement for Anderson and Fuller

I'm working through Rings and Categories of Modules by Anderson and Fuller. It's a great text, but I would like a more modern replacement. The text focuses too much on the language of generation and ...
2
votes
1answer
76 views

Number of ideals in a minimal irreducible decomposition

Assume $R$ is a local ring, $M\subseteq R$ is the maximal ideal, $I\subseteq R$ is an $M$-primary ideal and $I=\bigcap_{i=1}^n Q_i$ is a minimal irreducible decomposition of $I$ (i.e. $Q_i\subseteq R$ ...
4
votes
1answer
79 views

The realationship of $\hom(M\otimes_RN,M'\otimes_RN')$ and $\hom_R(M,M')\otimes\hom_R(N,N')$.

Let $R$ be a ring with identity, $M$ and $M'$ two right $R$-module, $N$ and $N'$ two left $R$-module. There is a natural way to define a homomorphism ...
0
votes
1answer
15 views

Module over different rings

If $M$ is a finitely generated abelian group, which can be made into a module over a ring $R$ (with a certain scalar multiplication) and has as a module the minimal spanning set $\{e_1, \ldots, ...
1
vote
2answers
75 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
47 views

Aluffi: submodule $\Longleftrightarrow$ cokernel?

Aluffi makes the following brief statement, in the context of modules: "The last sentence of Proposition 6.2 simply reiterates the slogan submodule $\Longleftrightarrow$ kernel and its mirror ...
1
vote
1answer
43 views

Exact sequence of modules and taking the quotient

Let $A$ be a commutative ring and $\text{Spec}\,A=\bigcup\limits_{i=1}^mD(f_i)$ be a covering by principal open sets. Show that the sequence of modules $$M\stackrel{\alpha}\to ...
2
votes
1answer
62 views

stationary set,club,module theory,Auslander lemma,

Here http://www.ams.org/journals/tran/1990-322-02/S0002-9947-1990-0974514-8/S0002-9947-1990-0974514-8.pdf I do not understand the first two lines of the proof of lemma 9,on page 550:What and why is ...
3
votes
2answers
87 views

Is there a relation between $End(M)$ and $M$ under tensor products?

Let $R$ be a commutative ring and $M$ be an $R$-module (If necessary, you can assume more conditions) Let $\phi$ be an $R$-endomorphism on $M$ and $\overline{\phi}:M^{\otimes n}\rightarrow M^{\otimes ...
0
votes
0answers
85 views

Relation between Tensor-hom adjunction and adjugate matrix

Let $R\to S$ be a ring homomorphism, let $M,N$ be $S$-modules and $Q$ an $R$-module. Then, we have $$\textrm{Hom}_R(M\otimes_S N,Q) \cong \textrm{Hom}_S(M,\textrm{Hom}_R(N,Q).$$ I want to know ...
2
votes
2answers
69 views

Why an R-module is an R/I-module precisely when it is annihilated by I?

Let $I$ be an ideal of $R$ and let $M$ be an $R$-module. Prove that the formula $(r + I) \cdot m = r \cdot m$ for $r + I \in R/I$, and $m \in M$ makes $M$ into an $R/I$-module if and only if $i ...
0
votes
0answers
35 views

The opposite of the right ideal in the ring of 2x2 matrices?

Since every ring has an opposite, I would like to know: Which is the opposite of the rings of $n \times n$ matrices? More specifically, of the $2 \times 2$ matrices. Is there an opposite for the ...
3
votes
2answers
88 views

For what kind of $R$-modules $M$ can we find an element $m\in M$ satisfing that $i:M\to M\otimes_R M, x\mapsto x\otimes m$ is an epimorphism?

Let $R$ be a commutative ring with identity and $M$ a $R$-module. I'm interested in under what condition we can find an element $m\in M$ satisfing that $i:M\to M\otimes_R M, x\mapsto x\otimes m$ is an ...
1
vote
2answers
32 views

Why this set is a $R/\mathfrak m$-module?

I'm reading this PDF: Discrete Valuation Rings and Function Fields of Curves. I'm trying to understand in this theorem why $\mathfrak m/\mathfrak m^2$ is a $R/\mathfrak m$-module (see number 4 below). ...
4
votes
1answer
201 views

Yoneda implies $\text{Hom}(X,Z)\cong \text{Hom(}Y,Z)\Rightarrow X\cong Y$?

I came across a result on the page Theorems implied by Yoneda's lemma? which said that Yoneda's Lemma implies the isomorphism of the title; namely, if we have $\text{Hom}(X,Z)\cong ...
10
votes
1answer
118 views

For abelian groups: does knowing $\text{Hom}(X,Z)$ for all $Z$ suffice to determine $X$?

Let $X$ and $Y$ be abelian groups. Suppose $\text{Hom}(X,Z)\cong \text{Hom}(Y,Z)$ for all abelian groups $Z$. Does it follow that $X \cong Y$? It has been answered before that this is true if the ...
3
votes
1answer
21 views

Presented matrix and number ring.

Let $V$ be the module generated by the column matrix $A= (2, 1+ \sqrt{-5})^T$. Prove that the residue of $A$ in $\mathbb{Z}[\sqrt{-5}]/ \mathfrak{P}$ has rank $1$ for every prime ideal ...
5
votes
2answers
85 views

What are the direct summands in $\mathbb{Z}^n$?

I am interested in knowing the rank $k$ submodules of the $\mathbb{Z}$-module $\mathbb{Z}^n$ which are are also direct summands. I know that $\mathbb{Z}^k$ sitting in $\mathbb{Z}^n$ in the standard ...
0
votes
1answer
54 views

Book for Module Theory

I want a book to cover the following topics in Module Theory: Modules ,Submodules,Quotient modules,Morphisms Exact sequences ,three lemma,four lemma,five lemma,Product and Co products,Free modules, ...
3
votes
1answer
36 views

Invertible matrix over a ring and its eigenvalues

Eigenvalues and invertible matrices for fields and vector spaces: Let $K$ be a field (so $K^n$ is a $K$-vector space) and let $A \in K^{n\times n}$ be an $n\times n$-matrix. Then we have the following ...
1
vote
2answers
44 views

If $M\bigoplus N $ submodule $A\bigoplus B$ does it imply either $M$ submodule $A$ or $M$ submodule $ B$?

If $M\bigoplus N$ is a submodule of $A\bigoplus B$ does it necessarily imply either $M$ is a submodule of $A$ or $M$ is a submodule of $B$? If in general it is not true, is it true if $M$ and ...
1
vote
2answers
54 views

A basic isomorphism of modules (useful for Corollary 2.7 of Atiyah and MacDonald).

Let $M$ be an $A$-module, $N$ a submodule of $M$, $\mathfrak{a} \subseteq A$ an ideal such that $M = \mathfrak{a}M + N$. Then $\mathfrak{a}(M/N) = (\mathfrak{a}M+N)/N$ I am having troubles in ...
2
votes
1answer
35 views

Singular matrix with entries in a ring. [duplicate]

Given a matrix $M\in A^{n\times n}$, where $A$ is a commutative ring different from $\{0\}$, then we know that if there exists a vector $x\in A^n$ such that $Mx=0$, then $\det M$ must be a zero ...
0
votes
2answers
55 views

tensor product of polynomial algebra

Is $R[x] \otimes R[x]$ a free $R \otimes R$-module? Here $R$ is a $k$-algebra and $\otimes = \otimes_k$.
9
votes
1answer
59 views

$M_1 \to M_2 \to M_3 \to 0$ exact iff $0 \to \text{Hom}_A(M_3, N) \to \text{Hom}_A(M_2, N) \to \text{Hom}_A(M_1, N)$ is exact. [duplicate]

Let $M_1 \to M_2 \to M_3 \to 0$ be a sequence of homomorphisms of $A$-modules. Is this sequence exact if and only if the induced sequence of abelian groups$$0 \to \text{Hom}_A(M_3, N) \to ...
3
votes
1answer
71 views

Coker of powers of an endomorphism

Let $F\in\operatorname{End}_R(M)$, where $M$ is a Noetherian $R$-module. If $\operatorname{Coker}F$ is of finite length, is Coker and Ker of all powers of $F$ of finite length? Is the condition of ...
3
votes
2answers
82 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the matrix ring $M_2(\mathbb{C})$. Let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional vectors). My ...
12
votes
2answers
77 views

What is $\mathbb{Z}[i] \otimes_{\mathbb{Z}[2i]} \mathbb{Z}[i]$?

What is $\mathbb{Z}[i] \otimes_{\mathbb{Z}[2i]} \mathbb{Z}[i]$? Also, since $\mathbb{Z}[i]$ is a PID, we should be able to write this $\mathbb{Z}[i]$-module as a direct sum of cyclic ...