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

learn more… | top users | synonyms

1
vote
1answer
162 views

Can one prove the existence of tensor product without explicitly constructing it? [duplicate]

R is a ring with 1. We construct tensor product $M \otimes N$ of right R-module $M$ and left R-module $N$ to basically be able to state its universal property that any R-bilinear map from $M\times N$...
0
votes
1answer
53 views

An element annihilates a module

Let $R$ be a ring, $A$ an $R$-module, $y$ belong in $R$ such that $1+y$ annihilates $A$. Then for any ideal $I$ containing $y$, prove that $IA=A$.
0
votes
1answer
37 views

Short Exact Sequences and R modules

Let $0 → A → B → C → 0$ be a short exact sequence of $R$-modules. Prove that for any $R$-module $M$ , there is a short exact sequence $0 → A \oplus M → B \oplus M → C → 0$. Can anyone please help me ...
2
votes
2answers
100 views

Flatness of $\mathbb{Z}$-modules

I have to prove that : 1) For every positive integer $n$, $\mathbb{Z}_{n}$ is not a flat $\mathbb{Z}$-module 2) Every torsion free abelian group is a flat $\mathbb{Z}$-module What I have done: 1) ...
1
vote
1answer
32 views

$M/im(f)$ is Torsion

Let $R$ be a Principal ideal domain and $f:R^{n}\rightarrow M$, be an injective homomorphism where $M$ is a finitely generated $R$-module. I need to show that $M/im(f)$ is Torsion. I know that since ...
1
vote
0answers
42 views

Change of basis - free submodule

Let $R$ be a commutative ring and $(e_1, \dotsc, e_n)$ be a basis for $R^n$, the free $R$-module of rank n. Let $A$ be an $n \times n$ matrix with entries in $R$. Let $f_i = \sum \limits_{j=1}^n a_{ij}...
0
votes
2answers
59 views

The meaning of functor $M \mapsto \mbox{Hom}_A(P,M)$ being exact

(I'm currently studying Lang's text Algebra and it comes up on the page 137. Lang does not explicitly define this expression.) Is the following understanding correct? The function $M \mapsto \mbox{...
4
votes
1answer
132 views

When is $N\otimes_A B \to N$ an isomorphism?

Let $A, B$ be commutative (unital) rings and $f\colon A \to B$ an $A$-algebra. There then exists a canonical functor $f_*\colon \mathbf{Mod}_B \to \mathbf{Mod}_A$ such that, for every morphism of $B$-...
0
votes
1answer
146 views

A submodule of a free module over a PID

$R$ is a principal ideal domain and $F$ is a free module over $R$ of infinite rank with basis $\{e_1,...,e_n,...\}$. Is it true that the $R$-submodule of $F$ spanned by $\{e_1,...,e_n\}$ has a ...
3
votes
0answers
170 views

Projective and simple modules over finite dimensional algebras

I'm working through some lecture notes on the representation theory of a finite dimensional algebra $A$ (associative, unital, over an algebraically closed field $k$), and have got stuck on a ...
2
votes
2answers
142 views

The Relationship Between Cohomological Dimension and Support

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as $$ \operatorname{cd}(I,M) \stackrel{...
2
votes
1answer
227 views

Do covariant functors preserve direct sums?

Suppose $T$ is a covariant functor from the category $R$-mod to Ab (the category of abelian groups) Is is necessary that $T(B \oplus C) \cong T(B) \oplus T(C) $ ? Does the answer change if we ...
0
votes
1answer
36 views

Extending morphism between derived functors

Let N,M be objects in a left exact functor $F:A\rightarrow B$ between abelian categoire's source, most importantly say there is an isomorpihsm $\psi: F(M)\rightarrow R^dF(N)$ is it possible to extend ...
0
votes
1answer
42 views

Equivalence between exact sequence of module and its induced one.

Let $X,X',X''$ be $A$-modules and denote by $\mbox{Hom}_A(X',X)$ the set of $A$-homomorphisms of $X'$ into $X$. Proposition 2.1 in the Lang's Algebra text states the following: A sequence $$ X' \...
1
vote
5answers
72 views

$|x| + |x-1| = 3$ how come its cases?

$$|x| + |x-1| = 3$$ in my textbook, they say that for this equation, there are 3 cases: $x\geq1$, $0 \leq x < 1$ and $ x < 0$ where do these come from and why? i thought, there are 4 cases ...
0
votes
1answer
89 views

Associated Graded Module

I'm learning about Associated Graded Ring and Associated Graded Module. I got definition from Wiki: My questions: What's $I^nM$?, it's mean: $I^nM=am, a\in I, m\in M$? With above define, so: $gr_I(...
0
votes
1answer
30 views

Property of modulo congruation

If I have: $$a^b \equiv 1 \mod xy$$ where $x,y$ are primes, is then true that: $$ a^b \equiv 1 \mod x$$ $$ a^b \equiv 1 \mod y$$ I don't sure if this is true, because I don't know how can I prove it
1
vote
1answer
68 views

Choice of ideal so that localisation of a module must be free?

If $R$ is a Noetherian (commutative) ring and $M$ is a finitely generated $R$-module, then what choice of nonzero ideal $I$ of $R$ is such that $M_P$ is a free $R_P$-module for any prime $P$ in $R$ ...
3
votes
1answer
64 views

Inequality amongst projective dimensions!?

Assume $φ : R\to S$ is a ring homomorphism between commutative rings sending unity to unity. Taking any $S$-module $M$ as an $R$-module, is it true that always $$\operatorname{pd}_R (M)\le \...
0
votes
2answers
67 views

Zero direct summand

I want a suggestion for the following: Let $M$ be a finitely generated $R$-module, where $R$ is a commutative Noetherian local ring with $1_R$, and $N$ a direct summand of $M$ such that $N⊆mM$, ...
1
vote
1answer
99 views

Projective dimension zero or infinity

Let $R$ be a commutative Noetherian local ring (having unity) with the maximal ideal $m$, which consists only of zero-divisors. Then for any finitely generated $R$-module $M$, the projective dimension ...
3
votes
1answer
123 views

Is this element necessarily non-zero in the tensor product?

Suppose $R$ is an integral domain and $M$, $N$ are torsion-free $R$-modules. If $m$ and $n$ are nonzero elements of $M$ and $N$ respectively then does it follow that $m \otimes n \neq 0$ ? If either $...
3
votes
1answer
89 views

Analogue of Baer criterion for testing projectiveness of modules

In order to test injectivity of a module $M$ it suffices to check if every linear map from an arbitrary ideal extends to the ring or not. Similarly in order to check the flatness of a module $M$ it ...
1
vote
1answer
79 views

Ultrafilter Lemma and Dimension Theorem

Reading on Wikipedia I find out that (the uniqueness in) the Dimension Theorem for arbitrary Vector Spaces can be proved using just the Ultrafilter Lemma (a strictly weaker version of Axiom of Choice)....
2
votes
1answer
66 views

$M \otimes_Z N =0$ if and only if $M \otimes_R N =0$?

If $R$ is a commutative ring, and $M$ and $N$ are $R$-modules, is $M \otimes_Z N =0$ if and only if $M \otimes_R N =0$? The forward direction seems clear. The backward direction seems like it should ...
3
votes
3answers
96 views

How to show $A\simeq \mathbb Z^{k-1}$?

I know this might be a silly question, but I don't know module theory very well. Suppose I have an exact sequence $$0\longrightarrow A\longrightarrow \mathbb Z^k\longrightarrow \mathbb Z\...
6
votes
2answers
471 views

Exercise 2.27 Atiyah-Macdonald, absolute flatness

A commutative ring $R$ is absolutely flat if every $R$-module is flat. Prove that the following are equivalent: 1) $R$ is absolutely flat 2) Every principal ideal of $R$ is idempotent 3) Every ...
2
votes
1answer
38 views

Inducing a module homomorphism

Suppose that I have a homomorphism of $A$-modules $M \xrightarrow{f} N$ ($A$ commutative with 1), a ring homomorphism $A \xrightarrow{g} A'$, and two $A'$-modules $M'$ and $N'$. Suppose further that ...
1
vote
1answer
76 views

Limit of inverse system where morphisms are non-zero products

I have a rather complicated inverse limit which I am trying to compute. I will try to distil the question to its barest form, but if more details are necessary I can supply them. Let $R$ be an ...
2
votes
0answers
56 views

Krull dimension of a direct limit of modules

Suppose that $\left\{M_{\lambda}\right\}$ is a directed system of $R$-modules, all of them with finite Krull dimension (specifically, all of them of dimension $n$). Is it true that $\dim\varinjlim M_{\...
1
vote
0answers
113 views

Resolution of module over polynomial ring

The problem is: Let $F$ be a field, and let $R = F[x_1, \ldots, x_r]$, the polynomial ring over $F$. Consider the $R$-module $M = R/(x_1, \ldots, x_r) \cong F$. Find a resolution of $M$ by free $R$-...
2
votes
1answer
92 views

Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and $q=p^r=3^2$....
1
vote
1answer
77 views

$\mathrm{End}_R(R[x])$ and $\mathrm{Hom}_R(R[x],R)$

Is $\mathrm{End}_R(R[x])$ (the set of all $R$-module homomorphism from $R[x]$ to $R[x]$) isomorphic to $R[x]$, for some ring $R$? Also, Is $\mathrm{Hom}(R[x],R)$ (the set of all $R$-module ...
7
votes
1answer
112 views

Describe all the quotient groups of $\mathbb{Z} \oplus \mathbb{Z}$

I'm thinking about a problem in algebraic topology of how to determine all the $k$-fold covers of $\mathbb{T}^2$, where $k$ is a certain integer. In thinking about the problem, I came upon the ...
1
vote
1answer
108 views

The existence of a projective resolution of M from finite rank free modules

Any R-module admits a projective resolution. When the ring is Noetherian, can we show the existence of a projective resolution of any finitely generated R-module by finite rank free R-modules? In ...
3
votes
0answers
81 views

Modules with finite injective dimension have $\omega_R$-resolutions

Let $(R,m,k)$ be a local Noetherian ring, $M$ a finitely generated $R$-module. How can I prove that $M$ has finite injective dimension if and only if it has a $\omega_R$-resolution? ($\omega_R$ is the ...
1
vote
0answers
49 views

Classify all possible $R$-module structures on a vector space

Let $V$ be an $n$-dimensional complex vector space. In particular, it is an Abelian group. Let $R$ be a (commutative, unitary) $\mathbb C$-algebra. Problem. I would like to parameterize all $R$-...
3
votes
1answer
191 views

When is a ring homomorphism also a R-module homomorphism?

Given a ring $R$(with or without $1$,may or may not be commutative), $R$ is also an $R-$module. Let $\phi_{ring}\colon R \to R $ be a ring homomorphism and $\phi_{module} \colon R \to R$ be a left $R-...
4
votes
1answer
177 views

Do these facts about $\mathbb{Q}/\mathbb{Z}$ generalize to more general cases?

We know that $\mathbb{Q}/\mathbb{Z} $ is injective. Not only that $0 \to A \to B \to C \to 0$ is exact iff $0 \to Hom(C,\mathbb{Q}/\mathbb{Z}) \to Hom(B,\mathbb{Q}/\mathbb{Z})\to Hom(A,\mathbb{Q}/\...
1
vote
2answers
96 views

Does tensoring with a module preserve this injection even if the module is not flat?

Suppose $R$ is a domain and let $Q$ be its field of fractions. Then $ 0 \to R \to Q $ is exact. Now suppose that $M$ is a torsion free $R$ module. Is it necessary that $ 0 \to M \to Q \otimes M $ ...
3
votes
2answers
70 views

Question about the set of all $\mathfrak p\in\operatorname{Spec}R$ such that $M_{\mathfrak p}=0$.

Is the following subset of $\operatorname{Spec}(R)$ always open? $R$ is a commutative, unitary ring and $M$ is an $R$-module.$$\{\mathfrak p \in\operatorname{Spec}(R) \colon M_{\mathfrak p} = 0\}$$
2
votes
1answer
501 views

Simple module over matrix rings

I'm trying to prove that if $M$ is a simple module over $M_n(D)$ where $D$ is a division algebra, then $M\cong D^n$. I know that if $M$ is a simple module over $R$ then it is isomorphic to $Rv=\{rv|r\...
2
votes
1answer
90 views

Is there an analogue of Bass-Papp theorem for Projective modules?

The Bass-Papp theorem for injective modules states that If $R$ is a commutative ring such that every direct sum of injective $R$ modules is injective then $R$ is Noetherian. Is there an ...
-1
votes
1answer
360 views

e * d = 1 mod phi(n); How do I find d? [closed]

given suppose e = 5, phi(n) = 96. How do i find the value of d? How do I solve this problem?
1
vote
1answer
32 views

Semi-simple and characteristic

Let $C_3$ be the cyclic group of order 3, then let $K$ be a field, let $X=span_K(1+c+c^2)$, show that $X$ has a complement if and only if characteristic of $k$ is not equal to $3$.
2
votes
1answer
132 views

Exercise 2.26 Atiyah-Macdonald, flatness

I'm stuck on this exercise. $A$ is a commutative ring with unit. $N$ is an $A$-module. Then $N$ is flat $\Longleftrightarrow $ $\text{Tor}_{1}(A/a, N ) = 0 $ for every finitely generated ideal $a$ of ...
2
votes
1answer
359 views

Endomorphisms of direct sum and division ring

How to prove that $$\operatorname{End}_R(V^{ \oplus n }) \cong M_n(D),$$ where $V$ is a simple left $R$-module and $D=\operatorname{End}_R(V)$. This is part of the proof finally leads to prove ...
3
votes
0answers
80 views

Approximating modules over complete local rings

Let $A$ be a complete local noetherian ring with maximal ideal $\mathfrak{m}$. Is the canonical functor $$\mathsf{Mod}(A) \to \varprojlim_n ~ \mathsf{Mod}(A/\mathfrak{m}^n),~ M \mapsto (M/\mathfrak{m}^...
1
vote
1answer
94 views

Short Exact Sequences

Let $M \ge N \ge P$ be R-modules. Prove that there exist natural (not depending on choices) R-homomorphisms $N/P \to M/P$ and $M/P \to M/N$ for which the sequence $0 \to N/P \to M/P \to M/N \to 0$ is ...
3
votes
1answer
72 views

Classify abelian groups $A$ which are irreducible $End(A)$-modules

Classify abelian groups $A$ which are irreducible $End(A)$-modules. I think i did it for finite abelian group $A$ . A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. ...