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

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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 ...
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1answer
21 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 = ...
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1answer
21 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 ...
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1answer
30 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 ...
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2answers
24 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 ...
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1answer
34 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 ...
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1answer
30 views

If N is a submodule of M, then λ(M) = λ(N) + λ(M/N).

Defining length of a module as follows, a module M has length λ(M) = n if there is a chain of submodules 0 = $M_{0}$ < $M_{1}$ < · · · < $M_{n}$ = M where n is maximal, and λ(M) = ∞ if there ...
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35 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 $N$ ...
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41 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 ...
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54 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?
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1answer
58 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.)
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36 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 $\varphi(m) \neq 0$?
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2answers
20 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 $C/L = B/A$?
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15 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?
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14 views

What are all the $k$ dimensional unimodular subspaces of $\mathbb{Z}^n$?

I am trying to prove the following assertion - The set of subgroups of $(S^1)^n$ which are isomorphic by an element of $Aut((S^1)^n)$ to the standard copy $(S^1)^k$ is naturally parametrized by the ...
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49 views

When does the duality functor commute with the wedge power functor?

When working with modules over a fixed commutative ring, I know that $(M \otimes N)^* \cong M^* \otimes N^*$ provided either $M$ or $N$ is finitely generated projective. Does it follow that ...
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1answer
19 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 ...
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2answers
47 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 ...
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45 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 ...
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1answer
29 views

Number rings as free module over base ring

Let $K \subset L$ be number fields and $\mathcal{O_K}, \mathcal{O_L}$ the corresponding rings of algebraic integers. Further let dimension$(L/K)$ = n as a vector space. If $K$ is a PID, then ...
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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 ...
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3answers
71 views

Finitely Presented Modules Definition

I am a little bit confused with the definition of finitely presented modules. In Lang's Algebra he defines a module $M$ to be finitely presented if and only if there is a exact sequence $F'\to F\to M ...
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2answers
54 views

Morphism of $k$-algebras between matrix rings over $k$

Let $k$ be a field and $f:M_n(k) \to M_m(k)$ be a morphism of $k$-algebras ($n,m \in \mathbb N$). Prove that $n$ divides $m$. I have no idea what to do here. I thought that maybe I should take an ...
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2answers
177 views

Vector space as simple $K[x]$-module

I am trying to solve the problem: Let $V$ be a vector space and $T$ a linear transformation $T:V \to V$. Let $(V,T)$ be a $K[x]$-module. Show that $(V,T)$ is simple if and only if $V$ is finite ...
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2k views

Examples of faithfully flat modules

I'm studying some results about flatness and faithful flatness and I'd like to keep in my mind some examples about faithfully flat modules. In general, free modules are the typical examples. ...
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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, ...
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1answer
56 views

Under what conditions does a quotient module $A/B\cong C$ imply the direct sum $A\cong B\oplus C$?

Suppose we have some finitely generated $R$-modules $A,\,B,\,C$ such that $A/B\cong C$. Under what conditions is it necessarily true that $A\cong B\oplus C$? Clearly the converse to this is true, but ...
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29 views

Conductor of a ring

An easy (possibly trivial) question from Neukirch's Algebraic Number Theory, p.47 Let $A$ be a Dedekind domain, $K$ its fraction field, $L$ a finite separable extension of $K$ and $B$ the integral ...
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24 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 ...
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36 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 ...
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1answer
71 views

Relation between two definitions of primary modules

Let $A$ be a commutative ring, $M$ be an $A$-module and $N \leq M$. There are two definitions of primary modules: 1) $M/N$ is coprimary (i.e., every zero divisor is nilpotent); 2) $\text{Ann}_A(N)$ ...
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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 ...
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30 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. ...
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1answer
11 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 ...
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27 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). ...
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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 & ...
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1answer
49 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 ...
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1answer
12 views

How do I calculate the dimension of the localization of a module?

Apologies in advance if this is a vague question. So I am trying to calculate the $\mathbf{Z}$-rank of a finitely generated torsion free module $M$ (I won't go into the details of what this module $M$ ...
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3answers
915 views

Learning to think categorically (localization of rings and modules)

I've been reading Ravi Vakil's notes on algebraic geometry notes and am having a hard time connecting the standard definition of the localization of a module to the definition in terms of universal ...
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1answer
39 views

Computing tensor products of $\mathbb{Z}$-modules.

I'd like to compute $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}^{n}$, for some natural number $n$, and $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$.
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1answer
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Length of quotients and relations between $\ell(\mathrm{coker}\varphi),\ell(R/\det\varphi)$

Let $R$ be domain(not necessary local) with maximal ideal $\mathfrak{p}$ and $d \in R, d \neq 0$. $(R/(d))_{\mathfrak{p}} = 0 \iff (d) \not\subset \mathfrak{p}$(?). And if $ (d) \subset \mathfrak{p}$ ...
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2answers
36 views

Identity of tensor products over an algebra

I hope this question hasn't been asked before, but I wasn't sure what to search so as to check. Suppose $R$ is a commutative ring with 1 and that $A$, $B$ are unital $R$-algebras. Suppose further that ...
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1answer
24 views

Factor Modules/Vector Spaces and its basis with canonical mappings

I'm having trouble with factor modules now. Well, specifically the following question from a past paper. Q. T=$R^2$ i.e. the real plane, and define $f:T$->$T$ with respect to the standard basis which ...
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183 views

Elementary divisors theorem for Dedekind domains (Exercise in Lang's Algebra)

Exercise 13 (b) of Chapter III in Lang's Algebra is as follows. Let $M$ be a finitely generated projective module over the Dedekind ring $\mathfrak{o}$. Then there exists free modules $F$ and ...
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1answer
12 views

Difference between an $R$-algebra being finitely generated and finite

So I have the two following definitions: An $R$-algebra $S$ is said to be finite over $R$ if it is finitely generated as an $R$-Module. An $R$-algebra $S$ is said to be finitely generated if $$S ...
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1answer
32 views

Existence of a Map

I just wanted to check to be sure I was correct as the more I stare at my proof the more I doubt myself. Suppose you have a commutative diagram of $R$-modules with exact rows: $$ ...
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2answers
34 views

A short question about the direct limits and direct sum of commutative rings.

By viewing a ring as a $\mathbb{Z}$-module, it is possible to define the direct limit of rings following the same procedure for modules. Let's suppose to work with commutative rings with unity. It ...
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43 views

What is the name of this special subset of module over commutative ring with unity

Let $R$ ba a commutative ring with unity and let $M$ be a $R$-module. For each $m,n\in M$ consider free spaces $\bar{m}$ and $\bar{n}$ generated respectively by $m$ and $n,$ i.e ...
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1answer
29 views

existence of module homomorphisms, so that a diagram becomes commutative

Let $R$ be a ring, $f: R^k \to R^m$ aswell as $g: R^l \to R^n$ module homomorphisms and $M = coker(f)$, $N = coker(g)$. Let $p_M: R^m \to M$ and $p_N: R^n \to N$ be the natural projections. I now ...
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2answers
26 views

properties of transformations inbetween quotient modules

Let $R$ be a ring, $M$ and $R$-left module and $U, V$ submodules of $M$. I want to show that $$f: M/U \cap V \to M/U \oplus M/V, m + U \cap V \mapsto (m + U, m + V)$$ and $$g: M/U \oplus M/V \to ...