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

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2
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1answer
38 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 ...
2
votes
1answer
22 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 $ ...
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 ...
4
votes
2answers
184 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
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
14 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
1answer
87 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 ...
2
votes
2answers
41 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 ...
0
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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
26 views

When is the external tensor product of indecomposable modules indecomposable?

Let $k$ a field and $A$, $B$ finite dimensional associative $k$-algebras. If $M$ is an irreducible $A$-module, and $N$ is an irreducible $B$-module, then $M\otimes_kN$ is an irreducible $A\otimes ...
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, ...
0
votes
1answer
18 views

Using Exchange Lemma in a decomposition

The "Exchange Lemma" in the decomposition of modules states that any decomposition of right $R$-modules $M_1⊕...⊕M_n=A⊕B$ with the endomorphism ring $End (A_R)$ local yields $M_i≈A⊕X$ for some $i$, ...
1
vote
1answer
44 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 ...
4
votes
1answer
80 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 ...
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). ...
0
votes
0answers
39 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 ...
3
votes
2answers
88 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 ...
3
votes
1answer
39 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 ...
0
votes
1answer
67 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, ...
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 ...
3
votes
1answer
45 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 ...
0
votes
0answers
86 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
1answer
36 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 ...
2
votes
1answer
78 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. $Q_i\subseteq R$ ...
12
votes
2answers
80 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 ...
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 ...
1
vote
1answer
21 views

$UTM_n[D]$ is artinian

Why is the upper triangular matrices over a division ring D is artinian? I tried to find properties of this class of rings. The only thing I found that the jacobson radical of this ring is the ...
1
vote
1answer
40 views

Free module over a set

I think I understand the definition of a free $R$-module over a set $X$: it is given by the set of all maps from $X$ to $R$ which vanish at all but finitely many points of $X$. The module operations ...
4
votes
2answers
86 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 ...
2
votes
1answer
39 views

Does “pseudo-independent implies independent” imply that $R$ is a field?

(All my rings are unital.) Suppose $R$ is a commutative ring and that $M$ is an $R$-module. Definition. Call a subset $X \subseteq M$ pseudo-independent iff for all proper subsets $Y$ of $X,$ the ...
2
votes
1answer
45 views

Perfect pairing induces isomorphism of tensor products

Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
0
votes
0answers
31 views

Is $\oplus_{i \in I} M_i$ necessarily projective?

If $\{M_i\}$ is a collection of projective modules over a ring $R$, then must the direct sum $\oplus_{i \in I} M_i$ necessarily be projective? I understand that each direct sum of two modules is ...
3
votes
1answer
30 views

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module?

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module? I understand I am supposed to think of $A$ as an $A \times B$-module by identifying ...
0
votes
0answers
19 views

Is there a natural link between symmetric polynomials and symmetric algebra?

Let $R$ be a commutative ring and $R[X_1,...,X_n]^{S_n}$ be the ring of symmetric polynomials. I have learned some basic properties of this ring and the results are really similar to those by ...
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 ...
0
votes
1answer
24 views

$\mathbb{Q}$ is a flat $\mathbb{Z}$-module

I am working on the following problem: Show that $0\rightarrow\mathbb{Q}\otimes A\rightarrow\mathbb{Q}\otimes B$ is exact where the map is $1\otimes i$ with $i$ being the inclusion map $A\subset B$. ...
1
vote
1answer
76 views

Exact functors preserve free modules?

Let $R$ be a principal ideal domain. Suppose that $F$ be an exact functor from the category of $R$-modules to itself. If $M$ is a free $R$-module, is $F(M)$ still a free $R$-module? I do not know how ...
0
votes
2answers
57 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 $\otimes = \otimes_k$.
0
votes
1answer
20 views

Property of free modules.

Let $R$ be a PID. I want to show that if $P$ is a finitely generated left $R$-module and $P$ is isomorphic to $R^n$ and $R^m$ (as $R$-modules) for $n,m$ in the natural numbers, then $n=m$. I was ...
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 ...
0
votes
1answer
34 views

Annihilators and exact sequence

Let $R$ be a commutative ring and $0\longrightarrow L \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}N\longrightarrow 0$ be an exact sequence of $R$-modules. How to prove ...
1
vote
1answer
37 views

A short exact sequence + exact sequence of opposite direction = split?

Let $0 \to A \to B \to C \to 0$ be a short exact sequence of modules over a commutative ring $R$ containing $1 \ne 0$. Suppose this is also another exact sequence $0 \to C \to B \to A \to 0$. Do ...
0
votes
1answer
36 views

$I$ is a two sided ideal in $A$, $L$ and $M$ are $A$-modules such that $L \subset M$. If $l\in L$ and $l\in MI$ then $l \in LI$?

$A$ is an Artinian ring, $I$ is a two sided ideal. I know that $L \subset M$ are $A$-modules, and that $l \in L \cap MI$. Is it true that $l \in LI$? If is not true, can I have an example where it ...
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 ...
4
votes
1answer
47 views

$M$ is a faithful $A$-module.

Let $A = M_2(\mathbb{C})$ and let $M = \mathbb{C}^2$ with its usual $A$-module structure. How do I show that $M$ is a faithful $A$-module? I understand that this amounts to showing that its ...
1
vote
3answers
58 views

Does there exist a module structure over $\mathbb{C}^2$ as a $\mathbb{C}[x]$-module?

Let $R = \mathbb{C}[x]$, $M = \mathbb{C}^2$. Does there exist a module structure over $M$ as an $R$-module or not?
2
votes
2answers
26 views

Faithful module is infinitely dimensional as a vector space.

Let $M$ be a $\mathbb{C}[x]$-module. Since $\mathbb{C} \subset \mathbb{C}[x]$, we may consider $M$ as a $\mathbb{C}$-vector space. If $M$ is faithful, must $M$ be infinite dimensional as a ...
0
votes
1answer
66 views

Equivalence between category of $R$-modules and $S$-modules

Is it true that by equivalence between category of $R$-modules and $S$-modules we conclude that $R$ and $S$ are isomorphic?($R$ and $S$ are unital rings.)