Tagged Questions

5
votes
1answer
41 views

a flatness criterion

I'm having trouble with part (b) of Exercise 10.5.25 from Dummit & Foote (the goal of the problem is to prove that $A$ is a flat $R$-module iff $A\otimes_R I\to A\otimes_R R$ is one-to-one for all ...
2
votes
1answer
48 views

$\mathrm{Hom}(R/I, R/J\otimes M)\cong ?$

Let $R$ be a Noetherian commutative ring, $I,J$ two ideal of $R$ and $M$ an $R-$module. Does anyone see the isomorphism $\mathrm{Hom}(R/I, R/J\otimes M)\cong \ldots$? Thanks.
4
votes
3answers
63 views

example of a flat but not faithfully flat ring extension

I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. ...
6
votes
1answer
70 views

is the dual of a finitely generated module finitely generated?

I recently thought of this and have no idea whether over a general commutative ring the dual of a finitely generated module is finitely generated. This must be known.
3
votes
1answer
41 views

Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
4
votes
0answers
54 views

Artinian rings are perfect

Is there a simple way to prove that an Artinian ring is perfect? (in the commutative case)
8
votes
2answers
55 views

Why over $\mathbb{Z}/n\mathbb{Z}$ projectivity, injectivity and flatness coincide for cyclic modules?

Assume $R=\mathbb{Z}/n\mathbb{Z}$ ($n\neq0$) and let $M$ be a cyclic $R$-module. Could you tell me how to prove that $M$ is projective if and only if it is injective if and only if it is flat? And ...
5
votes
0answers
90 views

An example of a commutative ring in which every primary ideal is prime

It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
1
vote
1answer
42 views

Extension of homorphisms on a divisible R-module

Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...
1
vote
0answers
62 views

A counterexample for $\operatorname{Ass}(M_1+M_2)$ [duplicate]

$\newcommand{\Ass}{\operatorname{Ass}}$ Let $A$ be a Noetherian ring and let $M$ be an $A$-module. Suppose $M=M_{1}+M_{2}$, then we have $\Ass(M)\supset \Ass(M_{1})\cup \Ass(M_{2})$. What is an ...
1
vote
0answers
42 views

$0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ exact, $M''$ flat. Why is $M$ flat $\Leftrightarrow M'$ flat?

Let $A$ be a commutative ring with identity, let \begin{align} 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0 \end{align} be an exact sequence of $A$-modules, let $M''$ be flat. I want ...
1
vote
1answer
47 views

What's stronger: projective or locally free? flat or locally free?

maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each ...
8
votes
2answers
126 views

What about a module of rank $\frac{1}{2}$?

Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...
3
votes
1answer
43 views

Difficulty Understanding Primary Modules

I have read that any irreducible sub-module $I$ of a Noetherian module $M$ is primary. However if we let $M = \mathbb{Z}_8$ and $I = \mathbb{4Z}_8$ this isn't true, because $I$ is irreducible, and ...
1
vote
1answer
85 views

Artinian ring and faithful module of finite length

Let $A$ be a ring. How can I prove that: $A$ is an Artinian ring $\Leftrightarrow \exists$ a faithful $A$-module which is of finite length. I know that if a ring has a faithful $A$-module which ...
3
votes
1answer
43 views

$S^{-1}B$ and $T^{-1}B$ isomorphic for $T=f(S)$

Let $f:A\to B$ be a homomorphism of rings, $S$ be a multiplicatively closed subset of $A$ and $T=f(S)$. Then $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$-modules. First we define the ...
6
votes
1answer
58 views

Showing that a ring is Noetherian

I show the following: Let $R$ be a commutative ring with unity and $I \subseteq R$ an ideal. Prove: if $R/I$ is a Noetherian ring and $I/I^2$ is a finitely-generated $R$-module, then $R/I^n$ is a ...
0
votes
3answers
71 views

Integral Dependence & Finitely Generated Modules

How to prove $(3)\Rightarrow(1)$ of this theorem: Let $A\subseteq B$ be commutative rings. The following are equivalent: $(1)~~x\in B$ is integral over $A$; $(2)~A[x]$ is a finitely generated ...
2
votes
1answer
56 views

Noetherian and Artinian modules over subrings

I have a question about whether Noetherian-ness and Artinian-ness of modules are preserved under changes of the base ring. More precisely: Let $R$ be a commutative ring and $S \subseteq R$ a ...
2
votes
1answer
37 views

The local cohomology modules are Artinian

Let $(R,m,k)$ be Noetherian local ring and $M$ a finitely generated $R$-module. Lemma 3.5.4 of Bruns-Herzog states that the local cohomology modules $H^i_m(M)$ are Artinian and that this ...
2
votes
2answers
51 views

Atiyah and MacDonald, Proposition 2.4

Let $M$ be a finitely generated $R$-module, $\mathfrak a \lhd R$ an ideal and $\phi:M\to M$ an $R$-linear map such that $\phi(M)\subseteq \mathfrak a M$. Then $\phi^n+a_1 \phi^{n-1}+\cdots+a_n=0$. ...
2
votes
1answer
33 views

On regular elements and Maximal Cohen-Macaulay modules

I was reading theorem 3.3.3 in Bruns-Herzog: we have a Cohen-Macaulay local ring $(R,\mathfrak m,k)$, $C$ and $M$ are maximal Cohen-Macaulay modules. (Probably to solve my question some of these ...
0
votes
1answer
33 views

The number of generators of a submodule over a Principal Ideal Ring.

Can someone give me a hint in proving that if a module $M=\langle m_1,\dots,m_n\rangle$ is generated by $n$ elements over a principal ideal (commutative) ring, then every submodule can be generated by ...
3
votes
1answer
43 views

Correspondence between submodules and quotient modules

What is the (natural) bijection between the set of all sub modules upto isomorphism and set of all isomorphic quotient modules upto isomorphism of a finitely generated torsion module over a PID. Is ...
3
votes
1answer
97 views

Question about isomorphism of modules.

I have been reading the book of DeMeyer and Ingraham "Separable Algebras of Commutative Rings," where in page 129 they prove the following. Let $\bullet$ Let $S$ be a commutative ring and $G$ be a ...
3
votes
2answers
74 views

How to prove that $\mathbb{R}$ is not artinian, considered as a $\mathbb{Q}$-module

How can I prove that the set $\mathbb{R}$ of real numbers as a $\mathbb{Q}$-module is not artinian? I have tried to prove it, by the irrationals, but I could not prove it.
1
vote
0answers
71 views

Injective hull commutes with Hom

Notation: $E_R(M)$ is the injective hull of $M$. Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ an $R$-module. Then $$\mathrm{Hom}_R(R/I, E_R(M)) \cong E_{R/I}(\mathrm{Hom}_R(R/I, ...
7
votes
1answer
77 views

Can we really understand $R$ by studying $R$-modules? [duplicate]

According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. Can ...
2
votes
2answers
36 views

Tests/ invariants for module isomorphisms

It two modules are indeed isomorphic, then it is often not too difficult to find an isomorphism since most of the time it is just the natural map. However, it takes some time for me to prove that two ...
4
votes
3answers
124 views

Right-adjoint functors are left-exact?

As a final exercise to VIII.1 in Algebra: Chapter 0, we are asked to prove If $\mathcal{F}\colon\operatorname{R-Mod}\to\operatorname{S-Mod}$ is a right-adjoint operator, then $\mathcal{F}$ is ...
1
vote
0answers
20 views

A property of linearly compact module

Let $(R,\mathfrak{m})$ be a noetherian local ring, $E$ the injective hull of $R/\mathfrak{m}$, $S=\operatorname{End}_R(E)$ and $M$ a linearly compact and discrete $R-$module. Show that ...
2
votes
1answer
49 views

Pure Submodules and Finitely Presented versus Finitely Generated Submodules

Let $A$ be a ring $M$ an $A$-module and $N$ a submodule. Definition: $N$ is called a pure submodule of $M$ if the sequence $0 \rightarrow N \otimes E \rightarrow M \otimes E$ is exact for every ...
3
votes
1answer
64 views

Finitely generated torsion module over a PID.

Let $A$ be a PID, $K$ be the field of fractions of $A$, and $M$ be a finitely generated torsion $A$-module. Let $M'=\text{Hom}(M,K/A)$ and $M''=\text{Hom}(M',K/A)$. I want to show that the evaluation ...
0
votes
1answer
41 views

Kernel of canonical morphism in inductive limit (proof by induction)

Let $\langle I, \leqslant \rangle$ be a directed poset and $\langle M_i, \mu_{i,j} \rangle$ be a directed system of $A$-modules over $I$. Now let $$ C = \bigoplus\limits_{i \in I} M_{i}, $$ and $D$ ...
5
votes
0answers
58 views

Injective Hull and and some Hom set.

Let $R$ be a commutative ring with unit. Suppose $P\in Spec(R)$ and let $E=E(R/P)$ be the injective hull of $R/P$. What can we say about $Hom_R(R/P, E)$. We know that $R/m\cong Hom_R(R/m, E)$, where ...
6
votes
2answers
133 views

Submodules of a free module over a commutative ring

Let $R$ be a commutative unital ring, $I$ a set, and $R^{(I)}$ the free module on $I$. Can there be a submodule $R^{(J)}\cong M\leq R^{(I)}$ with $|J|\!>\!|I|$? Can $R^{(I)}$ be generated (as a ...
1
vote
1answer
76 views

Direct Sum/Product of Flat Modules

Let $\left\{M_{\lambda}\right\}$ be a family of flat $A$-modules and define $M = \bigoplus_{\lambda} M_{\lambda}$. Let $0 \rightarrow N' \rightarrow N$ be an exact sequence of $A$-modules. Tensoring ...
2
votes
1answer
61 views

When is a quotient of an $R$-module $E$ a submodule of $E$?

Let $R$ be a commutative ring with $1$. Suppose we are given a surjective $R$-module map $\varphi:E \to M \simeq E/N$. Are there any sufficient and/or necessary conditions for having an injective map ...
2
votes
1answer
96 views

$M$ is $\bigcap \operatorname{Ass}(M)$-primary

Let $R$ be noetherian ring and $M$ an $R$-module such that $\operatorname{Ass}(M)$ is a finite set. Prove that $M$ is $\mathfrak{b}$-primary, where $\mathfrak{b}=\bigcap ...
2
votes
2answers
66 views

Maximal ideals generate maximal submodules?

Let $\mathfrak m$ be a maximal ideal of $R$ and $M$ an $R$-module such that $\mathfrak mM\ne M$. Is it true that $\mathfrak mM$ is a maximal submodule of $M$? Thank you. (I can see this happen in ...
-7
votes
1answer
133 views

Show that $T (S^{−1}M)= S^{−1}(T (M))$ [closed]

Let $R$ be an integral domain and $M$ an $R$-module. An element $m \in M$ is called a torsion element if $\operatorname{Ann_R(m)} \neq 0$, i.e. if $rm = 0$ for some nonzero $r \in R$. Denote the set ...
-4
votes
1answer
128 views

Show that $T(M)$ is a submodule of $M$.

Let $R$ be an integral domain and $M$ an $R$-module. An element $m \in M$ is called a torsion element if $\operatorname{Ann}_R(m) \neq 0$, i.e. if $rm =0$ for some nonzero $r\in R$. Denote the set of ...
1
vote
1answer
67 views

All associated primes appear in a series of submodules

This is essentially Ex VI.4.8 of Algebra: Chapter 0. Let $R$ be a commutative ring and $M$ be an $R$-module. Define \begin{equation} \operatorname{Ann}_{R}(m)=\{r\in R:rm=0\} \end{equation} for each ...
0
votes
1answer
302 views

The (Jacobson) radical of modules over commutative rings

Let $M$ be a module over a commutative ring $R$. Let $\Omega$ be the set of all maximal ideals of $R$. Prove that $\operatorname{Rad}(M)=\bigcap_{\mathfrak m\in \Omega}\mathfrak mM$, where ...
0
votes
1answer
57 views

Isomorphic completed modules, that were not isomorphic before completion

Let $M$ and $N$ be $R$-modules. Suppose we complete them with respect to an ideal $\frak{m}$ of $R$. If we have $$M^\wedge_\mathfrak{m} \simeq N^\wedge_\mathfrak{m}$$must if be the case that $M ...
3
votes
1answer
78 views

vector bundles on the affine line over a PID

Let $R$ be a PID. Is every finitely generated projective $R[T]$-module free? In other words, is every vector bundle on $\mathbb{A}^1_R$ trivial? For $R=k[X]$ this is true by the Theorem of ...
0
votes
0answers
56 views

Does finitely generated associated graded module imply stable filtration?

Let $R$ be a Noetherian ring and $\mathfrak a$ be an ideal of $R$. Then (i) $G_{\mathfrak a}(R)$ is Noetherian. (ii) If $M$ is a finitely generated $R$-module and $\mathcal F=\{M_n\}$ is a stable ...
6
votes
2answers
128 views

Restriction of scalars and tensor product

All rings I'll consider will be commutative with identity. Given a homomorphism $f:R \to S$ we can give an $S$-module an $R$-module structure via restriction of scalars. In particular, $S$ can be ...
2
votes
0answers
64 views

Separability of finitely generated projectives over commutative ring

A class $\mathcal{C}$ of $R$-modules is called -separative if $A \oplus A \simeq A \oplus B \simeq B \oplus B$ implies $A \simeq B$ for each $A,B \in \mathcal{C}$ -cancelative if $A \oplus C \simeq ...
0
votes
0answers
51 views

The $I$-torsion submodules of an injective module [duplicate]

Possible Duplicate: Prove that the following module is injective Prove that $I$-torsion submodules of injective modules are injective (the ring is Noetherian). Please help me.

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