3
votes
0answers
51 views

Automorphism of certain f.g. free modules

This is a quick question from Frohlich and Taylor's Algebraic Number Theory, II.4, p 94. Let $R$ be a Dedekind domain with quotient field $K$, $\mathfrak p$ is a non-zero prime ideal of $R$ and ...
1
vote
1answer
27 views

Basis for the completion of a free module

This (or similar) question might have been asked before- apologies for any duplication. I've got a Dedekind domain $R$, a non-zero prime ideal $P$ of $R$ and the completion $\widehat{R}$ of $R$ wrt ...
2
votes
1answer
38 views

How can I prove commutative rings are stably finite?

I have a proof that proves that a stably finite ring (i.e. one where $AB=I\iff BA=I$ for any two square matrices $A,B$ with entries in the ring) has the Invariant Basis Number (IBN). Trouble is, it is ...
0
votes
1answer
30 views

tensor product of R-algebra and f.g module [closed]

$R$ is a commutative noetherian ring. If $S$ is an $R$-algebra, and $M$ a finitely generated $R$-module, is $M\otimes_RS$ finitely generated $S$-module? If this is not true, what can i add to ...
1
vote
1answer
42 views

exact sequence problem

Let $B_1 \stackrel{f}{\to} B \stackrel{g}{\to} B_2 \to 0$ be an exact sequence. For any module $M$ the sequence $$0 \to \mbox{Hom}(B_2,M) \stackrel{g*}{\to} \mbox{Hom}(B,M) \stackrel{f*}{\to} ...
3
votes
1answer
46 views

Maximal linearly independent sets in a f.g. module

Suppose $M$ is a finitely generated module over a commutative unital ring $R$. Is it true that every maximal linearly independent set in $M$ has the same size? What is the most general condition ...
4
votes
1answer
58 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 ...
2
votes
1answer
20 views

Show that if $R$ is principal, $N $ is pure and $Ann(x+N)= Rd$ then there exists $y \in M$ such that $x+N=y+N$ and $Ann(y)=Rd$

Let $R$ a integral domain and $M$ a $R$-module. A submodule $N$ of $M$ is pure if for all $x \in M$ and $a \in R$ such that $ax \in N$, there exists $y \in N$ such that $ax=ay$. Show that if $R$ ...
2
votes
1answer
36 views

Basis of a free module is the localization of its basis?

Let $R$ be a ring and $M$ a free $R$-module with basis $B$. If $P$ is a prime ideal of $R$, is $B_P$ a basis for the $R_P$-module $M_P$?
2
votes
1answer
65 views

Show that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic to the field $\mathbb{Q}$

I have problem in showing that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic as a ring to the field $\mathbb{Q}$. Any idea? Thanks
1
vote
0answers
20 views

Monotonic Union of Submodule

The theorem is: Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule. Is ...
5
votes
1answer
87 views

On the indecomposable decomposition of the reduction of an integral representation

Here is a problem I have been grappling with all day. I started out thinking it might be true but am now inclined to believe it is false, and would like to see a counterexample. Suppose $G$ is a ...
2
votes
2answers
115 views

Applications of groups, rings and modules in life [closed]

I have read calculus. Now I'm reading algebra. What is the application of groups, rings and modules in life? When we study derivations we know several applications. What about groups, rings and ...
1
vote
1answer
25 views

Show that if $M$ is a R-module free, for all $r \in R$ and $m \in M$ such that $r \neq 0$ e $m \neq 0$ we have $rm \neq 0$.

is it true that if $M$ is a $R$-module free, for all $r \in R$ and $m \in M$ such that $r \neq 0$ e $m \neq 0$ we have $rm \neq 0$. Comments: I tried to do the following: Suppose that $rm = ...
3
votes
1answer
24 views

Minimal set of generators VS Length of a module.

Recently, I have been thinking about a problem in which I try to use the information I have about the set of generators of a finitely generated module over a nice ring and say something about the ...
2
votes
1answer
60 views

Jacobson radical of a ring finitely generated over $\mathbb Z$

If a commutative ring $R$ with $1$ is finitely generated over $\mathbb Z$ could one deduce that the Jacobson radical of $R$ is nilpotent? I am aware of the well-known fact that when $R$ is ...
1
vote
0answers
27 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and ...
2
votes
1answer
63 views

Integral domain with a finitely generated non-zero injective module is a field

Suppose that $R$ is a integral domain. Suppose that there exists a non-zero finitely generated injective module $M$. How can I prove that $R$ is field?
3
votes
1answer
59 views

Direct product of Cohen-Macaulay rings/Eisenbud, Exercise 18.6

Somehow I believe (or doubt (!)) that direct product of two Cohen-Macaulay (C-M) rings may not be C-M. Can anybody give me an example verifying this? I would be grateful to him/her.
0
votes
0answers
51 views

Sufficient conditions for quotient ring to be Cohen-Macaulay

We know that every Noetherian integral domain with (Krull) dimension $1$ is Cohen-Macaulay (CM). In a commutative algebra text the author have presented the following problem: "Let $(R,m)$ be a CM ...
2
votes
1answer
25 views

Difference of a ring and its module over itself

What is the difference of a ring $R$ and the module $R_R$? It looks like they are just the same thing. Is it correct that the difference is $R$ is a ring with multiplication defined but $R_R$ is an ...
1
vote
1answer
23 views

Morita equivalence: Is $_{\mathrm{End}_R(P)}P$ projective if $P_R$ is?

Assume $P$ is a right projective $R$-module. Is $P$, viewed as a left $\mathrm{End}_R(P)$-module, projective as well? If not, under what conditions does it hold? Context: I am trying to ...
2
votes
1answer
26 views

Irredundant intersection of submodules

Let $A$ be a commutative ring, $M$ an $A$-module, and $N_\alpha\subset M$ a family of submodules. Consider the intersection $$\bigcap_\alpha N_\alpha.$$ We say that the intersection is irredundant if ...
2
votes
1answer
74 views

Quasi-hereditary algebras

Can anyone please recommend a reference (I prefer a book chapter) on Quasi-hereditary algebras? and if it is possible tell me how to prove that Schur algebras are Quasi-hereditary (or any other ...
0
votes
1answer
28 views

$\hom_R(A,B)$ is finitely generated if $R$ is noetherian [duplicate]

This is part of an exercise I'm doing, from Rotman Introduction to homological algebra. Let $R$ be a commutative ring, and let $A$ and $B$ be finitely generated $R$-modules. Then if $R$ is ...
2
votes
1answer
41 views

Starting projective modules problems

Show that if $n=rs$ where $n,r,s>1$ are positive integers, then the $ \mathbb{Z}_n $-module $r \mathbb{Z}_n$ is projective but it is not free if $(r,s)=1$. Any ideas or help how to prove this ...
2
votes
1answer
71 views

Newbie into categorical proofs

Let $$ F:Mod_A \to Mod_B $$ an aditive , exact and covariant functor and $$ M ∈ Mod_A $$ and $$M_1 , M_2 $$ submodules of M . Show that $$ F( M_1\cap M_2)=F(M_1)\cap F(M_2 ) $$ $$ F( M_1+ ...
0
votes
2answers
36 views

Not so usual equivalence of maximal left ideal of a ring

I was reading Foundations of Module and Ring Theory and i found this equivalence of maximal left ideal as exercise in the the first chapter: A left ideal $I$ of a ring $R$ is a maximal if and only ...
0
votes
1answer
26 views

Chains in modules

I'm answering this question: Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated but $A$ is finitely generated whenever ...
3
votes
1answer
36 views

$\bf{Z}$-module homomorphism and $\bf{Q}$-module homorphism

$R$-modules are also $\bf{Z}$-modules and $R$-module homomorphisms are also $\bf{Z}$-module homomorphisms. If $M$ and $N$ are $\bf{Q}$-modules and $f : M \rightarrow N$ is a $\bf{Z}$-module ...
0
votes
0answers
44 views

Minimal number of generators and associated prime ideals

Let $R$ be a commutative Noetherian local ring, $M$ a finitely generated $R$-module with minimal number of generators $\nu_{R}(M)=s$, and let $Q$ be the total quotient ring of $R$. Then, ...
1
vote
1answer
25 views

Is there a direct proof that $M_n(\mathcal k)$ is semisimple ring.

An R-module M is called semisimple if on of the following condition holds: 1) M is a direct sum of simple* submodules of M 2) M is a sum of simple submodules of M 3) For any R-submdoule N of M ...
1
vote
1answer
47 views

Symmetric powers of ideal quotients in a local ring.

Let $R$ be a local ring and $I \subset R$ any ideal. When is it the case that $(I \: \backslash I^2)^n = I^n \: \backslash I^{n+1}$? Put another way, when is the natural map $\text{Sym}^n(I/I^2) ...
1
vote
1answer
55 views

A statement equivalent to flatness

If $R$ is a ring with identity and $P$ is a flat right $R$-module, it is a fact that any $R$-homomorphism $f$ from a finitely presented right $R$-module $M$ to $P$ factors through a finitely generated ...
3
votes
0answers
57 views

Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
0
votes
1answer
36 views

Maximal Ideals of Matrix Ring

Let $D$ be a division algebra. I'm trying to show that all simple $M_n(D)$ modules are isomorphic to $D^n$. I know that $M_n(D)=D^n\oplus D_n \oplus \dots \oplus D^n$ (n terms). I have that if $S$ is ...
0
votes
0answers
18 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 ...
0
votes
1answer
41 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 ...
0
votes
0answers
41 views

Example of a module over a non-abelian ring

Is there an example of a module over a non-commutative ring in which two maximal independet sets have distinct cardinality? Let $M$ be a module over a ring $R$. As usually a subset of ...
1
vote
1answer
33 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: ...
0
votes
2answers
39 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
72 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 ...
1
vote
1answer
55 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 ...
5
votes
2answers
300 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 ...
1
vote
0answers
54 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 ...
2
votes
0answers
39 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
29 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 ...
2
votes
1answer
38 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 ...
1
vote
1answer
29 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 ...
0
votes
1answer
28 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 $3$.