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

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4
votes
1answer
57 views

$2 \otimes_{R} 2 + x \otimes_{R} x$ is not a simple tensor in $I \otimes_R I$

This question is related to Properties of the element $2 \otimes_{R} x - x \otimes_{R} 2$. Another exercise of Dummit-Foote is to show that Let $I = (2, x)$ be the ideal generated by $2$ and $x$ ...
1
vote
1answer
59 views

$\biggl( \prod_{i=1}^{+ \infty} \mathbb{Z}/2^i \mathbb{Z} \biggr)\otimes_{\mathbb{Z}}\mathbb{Q} \neq 0$ without using $p$-adic integers

I'm doing this exercise from Dummit-Foote: Show that tensor products do not commute with direct products in general. [Consider the extension of scalars from $\mathbb{Z}$ to $\mathbb{Q}$ of the direct ...
2
votes
1answer
87 views

An equivalence for $\operatorname{grade}(I,M)\ge 2$. [duplicate]

Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Show that $\operatorname{grade}(I,M)\ge 2$ iff the canonical homomorphism $M \mapsto\operatorname{Hom}_R(I,M)$ is an ...
1
vote
1answer
258 views

On the grade of an ideal

I need to prove the following statment (actually a special case of it). Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Then $\operatorname{grade}(I,M)\geq 2$ if ...
0
votes
1answer
103 views

Show that if an ideal is free as a module then it is principal.

Here $\mathfrak a$ is an ideal of a commutative ring $A$. Show that $\mathfrak a$ is principal if it is free as an $A$-module.
3
votes
2answers
71 views

How can I compute $\operatorname{Tor}(\mathbb Z_{p},\mathbb Z_{q})$?

I am self-studying Vick's Homology Theory, and now it is on the topic of free resolutions. Since I am not familiar with it, I have little ideas about how to compute $$\operatorname{Tor}(\mathbb ...
3
votes
2answers
41 views

$M \otimes_R K$ is uniquely divisible as $R$-module

A module $M$ over a domain $R$ is called uniquely divisible if for every $r \in R \setminus\lbrace 0 \rbrace$ the endomorphism $$\phi_r : M \to M$$ $$m \mapsto r \cdot m$$ is a bijection. Let $K$ be ...
4
votes
1answer
51 views

$Q(\mathbb{Z}[t]) / \mathbb{Z}[t]$ is not injective

I am doing this exercise: Let $R = \mathbb{Z}[t]$ and let $K$ be its fraction field. Show that the $R$ module $K/R$ is divisible but not injective. I have done the divisible part, but I am stuck on ...
0
votes
0answers
30 views

Suggest a good book or reference on graded modules over polynomial rings

I am looking for reference books or papers on graded modules over the polynomial ring $k[x_0, \ldots, x_n]$. Any good commutative algebra text like Eisenbud's Commutative Algebra already contains a ...
1
vote
1answer
26 views

Dimension of a module (as vector space)

I am reading the proof of the following (part of a) lemma: Let $A$ be a K-algebra. If $A$ is finite, then any nonzero module contains a simple submodule. The proof of this is where I am stuck: ...
0
votes
0answers
53 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
20 views

Quick question: SES where base ring is a PID

Let $M$ be an $R$-module, where $R$ is a PID. Assume there is a free $R$-module $F$ and a surjective map $\phi :F\rightarrow M$. Then why is $\ker(\phi)$ also free? Thank you.
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
27 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 ...
0
votes
1answer
25 views

$Hom_{A-Alg}(Sym(M),B)\cong Hom_{A-mod}(M,B)$

Let A be an ring. M be an A-module. Let $Sym (M)$ be the Symmetric algebra of M over A. Let B be an A-algebra. Why is $Hom_{A-Alg}(Sym(M),B)\cong Hom_{A-mod}(M,B)$ One way is clear - If we have a ...
0
votes
0answers
118 views

When is $\operatorname{gr}_I (M)$ finite?

When is $\operatorname{gr}_I (R)$ (I mean associated graded ring of $I$) finite? When is $\operatorname{gr}_I (M)$ finite? ($R$ is Noetherian ring and $M$ finite $R$-module.)
1
vote
1answer
19 views

Characterization of faithfully flat modules

This is an exercise from Rotman, introduction to homological algebra. A right $R$-module $B$ is called faithfully flat if : 1) $B$ is flat 2) If $X$ is a left $R$-module and $B \otimes_R X =0 $ ...
1
vote
1answer
50 views

Pushout of an injective map is injective

This is an exercise from Rotman , Introduction to homological algebra. Given a pushout diagram in $R$-Mod $$\begin{array} AA & \stackrel{g}{\longrightarrow} & C \\ \downarrow{f} & & ...
2
votes
2answers
77 views

If $M_1\cap K=M_2\cap K$ and $M_1+K=M_2+K$ then $M_1=M_2$?

I have a ring $R$ with $K\le M$ and submodules $M_1,M_2$. If we have that: $$M_1\cap K=M_2\cap K \text{ and } M_1+K=M_2+K$$ can we conclude that $M_1=M_2$? I don't think that this is true ...
4
votes
1answer
41 views

Dual of Schanuel lemma

This is an exercise from Rotman, Introduction to homological algebra. Given exact sequences of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & M & \overset{i}{\longrightarrow} ...
0
votes
0answers
11 views

Verification of step in a proof of the decomposition of primary f.g torsion modules over PIDs?

I was reading about the decomposition of finitely generated primary torsion modules over PIDs, and though of an alternative way to do the "inductive" step. Since it is substantially simpler than the ...
0
votes
1answer
34 views

Applying an additive covariant functor to a sequence

Let $F$ be an additive covariant functor and $0→A_1→A_1⊕A_2→A_2→0$ be the usual exact sequence in the category of $R$-modules with the first map injection on the first, and the second map the ...
1
vote
0answers
37 views

Direct limits commute with $\mathrm{Tor}$ functor

How one could prove that direct limits commute with the functor $\mathrm{Tor}$? Of course, I know that $\mathrm{Tor}$ with its first $0$ index is the same as tensor product which does commute ...
0
votes
1answer
35 views

Proves or counterexamples in retraction and coretractions of modules

Any tip for proving or counterexamples that the following morphism of $\mathbb Z$-modules ${\mathbb Z} \to {\mathbb Q}$ is not a retraction and ${\mathbb Q} \to {\mathbb Q/\mathbb Z} $ is not a ...
0
votes
1answer
29 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
43 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
72 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+ ...
1
vote
2answers
75 views

Help proving a short exact sequence

Show the following sequence is an exact sequence of $\mathbb Z$-modules when $n$ is a positive integer such that $n=rs$: $$ 0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0. $$ ...
3
votes
1answer
116 views

Torsion-free and projective modules over a Dedekind domain

Suppose that $A$ is a Dedekind domain (and integral domain). I am trying to prove that if $M$ is a torsion-free $A$-module, then it is projective and vice versa. Suppose that $M$ is projective. Then ...
0
votes
1answer
29 views

Definition of direct summand

If M is an R-module ,and if N is a direct sum of M over an indexed set I, then is it correct to write: N= T direct sum M, where T is the direct sum of M over I-{i} , where i in I.
0
votes
1answer
39 views

Proving that the following is an exact sequence

If $L$ and $N$ are submodules of $M$ show that the following is an exact sequence: $$ 0\to M/(L\cap N) \to (M/L)\oplus (M/N) \to M/(L+N )\to 0 $$ for the last morphism $g(x +L , y+N) = (x-y) + ...
0
votes
1answer
32 views

Every finitely generated module is the quotient of a finitely generated projective module.

Every finitely generated module is the quotient of a finitely generated projective module. I already find a proof of this but it uses tensor functor and flat modules propositions so im looking for a ...
-1
votes
1answer
58 views

Basic short exact sequence exercise in $\mathbb{Z}$-modules [closed]

I was wondering how to properly argue the proof of the following: a) Prove that there is an exact sequence $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}_n \to 0$ (consider $f : \mathbb{Z} \to ...
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 ...
6
votes
1answer
392 views

Annihilator of quotient module M/IM

Let $A$ be a commutative ring, $I$ an ideal of $A$ and $M$ an module over $A$. Is it true that $\operatorname{Ann}(M/IM) = \operatorname{Ann}(M) + I$? One inclusion is certainly true, but I ...
0
votes
0answers
38 views

Cokernels of Modules and Exact Sequences

Suppose we have n-dimensional free modules over a dvr $T_1$, $T_2$, $T_3$ (i.e. $T_i \cong R^n$) such that $$0 \to T_1 \to T_2 \to T_3 \to 0$$ is exact. Suppose further that $E_1,E_2,E_3$ are ...
1
vote
0answers
23 views

Small submodule

Let $P$ be an $R$-module. If $P$ is projective (not neccesarily finitely generated) $R$-module and $top(P)$ is projective does this imply that $rad(P)$ is a small submodule of $P$? What if $top(P)$ is ...
0
votes
1answer
27 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 ...
5
votes
1answer
393 views

Showing the countable direct product of $\mathbb{Z}$ is not projective

I am trying to prove that the direct product $M = \mathbb{Z} \times \mathbb{Z}\times \cdots$ is not a projective $\mathbb Z$-module and I am stuck near the end of the proof because of the authors use ...
2
votes
1answer
24 views

Quotient different modules by the same module to get the same module.

More precisely, If i have modules $M$ and $M'$ and $N$ and $N'$ does $M/N \cong M'/N'$ and $N \cong N'$ imply that $M \cong M'$? What if we also know that $N$ is free or that $M/N$ is free?
0
votes
1answer
18 views

Minimal Frattini Extensions

From "Construction of Finite Groups" article : "To construct minimal Frattini extensions of $H$ we have to consider all suitable irreducible $H$-modules $M$. Clearly, $M$ can be viewed as an ...
1
vote
1answer
49 views

Cover of a direct summand

Let $L,N$ be $R$-modules. If $L$ and $L \oplus N$ have projective covers, is it true that $N$ admits a projective cover?
1
vote
0answers
28 views

If a direct sum has a projective cover, must the summands have projective covers?

In “Cover of a direct summand” it is asked to show that if a direct sum has a projective cover, and if one of the summands has a projective cover, then so does the other. I gave a solution that works ...
4
votes
1answer
281 views

A formula for the minimum number of generators of a module over a semilocal ring

Let $R$ be a commutative ring with only finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_r$. Let $M$ be a finitely generated $R$-module. Then $$\mu_R(M)=\max\{\dim_{R/\mathfrak ...
6
votes
2answers
223 views

How can I find an element $x\not\in\mathfrak mM_{\mathfrak m}$ for every maximal ideal $\mathfrak m$

Let $R$ be a commutative ring with finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_n$. Let $M$ be a finitely generated module. Then there exists an element $x\in M$ such that ...
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, ...
2
votes
1answer
39 views

$M_R$ is finitely generated iff Every submodule of $M_R$ is finitely generated

$M_R$ is finitely generated Every submodule of $M_R$ is finitely generated. Do the sentences above have the same meaning? Thanks for any replies.
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 ...
1
vote
1answer
54 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 $R$ ...