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

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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 ...
3
votes
1answer
34 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
46 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
41 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
82 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
32 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
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. $Q_i\subseteq R$ ...
12
votes
2answers
73 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
58 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
17 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
35 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 ...
3
votes
2answers
80 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
37 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
44 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 ...
1
vote
2answers
38 views

Cyclic projective module [closed]

Let $R$ be an integral domain. If $M$ is a cyclic $R$-module which is also projective, then must there necessarily be an isomorphism of $R$-modules $M \cong R$?
0
votes
0answers
28 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 ...
2
votes
1answer
27 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
17 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
53 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
22 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
66 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
52 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
37 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
33 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
34 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
45 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
57 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
25 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
64 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.)
5
votes
0answers
41 views

Exists an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?

Let $M$ be an $R$-module with some torsion element $m$. Does there exist an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?
0
votes
1answer
28 views

vector space constructed through a torsion module

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form: $$ M = R/(p^{e_1}) \oplus \dots \oplus R/(p^{e_t}). $$ Let now be $_pM = ...
0
votes
2answers
23 views

Isomorphism of subquotient and quotient modules

Is it true that for every submodules $A\subset B$ of $C$ there is some submodule $L\subset C$ such that $C/L = B/A$?
0
votes
0answers
16 views

Split sequence and complement of a submodule

I have proved the equivalence of 1 and 2, but have no idea about 3.Could anyone offer some advice?
1
vote
1answer
21 views

Does the restriction of scalars functor preserve the quotient module construction?

I have two rings $R,\,S$ (not necessarily commutative - in the case I have in mind, just $R$ is commutative) such that $i:R\subset S$. Clearly, if we have a module $M$ over $S$, then we can restrict ...
0
votes
0answers
48 views

Abstract interpretation of isomorphism between tensor product with dual and hom

I'm interested in the following statement, coming from Remark 6.4.21 of Qing Liu's Algebraic Geometry and Arithmetic Curves: Let $\mathcal{F}, \mathcal{G}$ be quasi-coherent sheaves on a scheme ...
1
vote
2answers
62 views

Is the minimal number of generators of an ideal the rank of the ideal as a free $\mathbb Z$-module?

In an algebraic number theory course, my lecturer said that any ideal of $\mathcal O_K$, where $K$ is a quadratic number field, is generated by at most two elements. I am wondering why this is. When ...
2
votes
1answer
20 views

Describing the Kernel of an identity-preserving ring morphism from a ring $R$ to an Endomorphism ring of an additive Abelian Group.

I'm currently working through TS Blyth's book on Module Theory (Module Theory: An Approach to Linear Algebra). From Exercise 2.3: "Let $M$ be an Abelian Additive Group, and $R$ a unitary ring. Let ...
0
votes
0answers
16 views

free submodules of free modules of a PID

Let $R$ be a principal ideal domain. Now I want to show that each submodule $N \subseteq M$ of a free $R$-module $M = R^{(I)}$ is also free. As a hint, it says I might consider the tripel $(J, ...
0
votes
2answers
40 views

Finitely generated modules, related to the Gaussian integers

Let $M = \mathbb{Z}[i] = \{a + bi|a, b \in \mathbb{Z}\}$ be the additive group of the Gaussian integers. Consider the $\mathbb{Z}$-submodules $N_1 = 2M$, $N_2 = (1 + i)M$. I now want to figure out ...
0
votes
0answers
32 views

The maps $\delta: V \to B \otimes V$ and $\delta': V \otimes V^* \to B$.

Let $B$ be a coalgebra and $V$ a vector space. Suppose that we have a coaction $\delta: V \to B \otimes V$. Is the map $\delta$ equivalent to a map $\delta': V \otimes V^* \to B$? Thank you very much. ...
1
vote
1answer
15 views

Can one extent to (left) modules on IBN ring the results of vector spaces?

We know that (left) modules "are not as nice as vector spaces" because lots of properties of these ones don't fit to the most general modules. But il we restrict to modules on IBN ring (i.e. modules ...
0
votes
0answers
27 views

properties of finitely generated torsion-modules and their submodules over a PID

Let $R$ be a principal ideal domain, $M$ a finitely generated $R$-torsion module, and $N \subseteq M$ a submodule. I want to show that there exist free R-modules $F, F', F''$ and module homomorphisms ...
0
votes
0answers
28 views

Does the dual behave well with base change?

Let $R$ be a ring, $M$ an $R$-module and $R \to S$ a ring homomorphism. I wondered under which conditions we have an isomorphism $$ Hom_R(M,R) \otimes_R S \xrightarrow{\sim} Hom_S(M \otimes_R S, S). ...
4
votes
0answers
61 views

If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?

Let $A$ be a local (not necessarily noetherian) ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Let $M$ be a finitely generated $A$-module such that $\mathfrak{m}\otimes_A M\rightarrow ...
1
vote
0answers
37 views

Need more insight on a formula

Following is a part of a programming contest problem. Given $C_{1},m,n,o,x,y,z,c,d,K,J$ are positive integers $ C_{i} = \left\{ \begin{array}{l l} (m*C_{i-1}^2 + n*C_{i-1} + o) \bmod J & ...
2
votes
1answer
52 views

When is composition in $A$-mod surjective?

Let $k$ be any field. Suppose that $A$ is a finite dimensional associative $k$-algebra. Let $A$-mod be the category of finite dimensional (over $k$) left $A$-modules. Then composition of ...