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

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1answer
38 views

Show that $\operatorname{Hom}_R(E, F)$ is an $R$-module in a natural way.

Let $R$ be a conmutative ring, and $E,F$ are modules, show that $\operatorname{Hom}_R(E, F)$ is an $R$-module in a natural way. Is this still true if $R$ is not conmutative? I think in the to ...
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1answer
35 views

The Chinese remainder theorem for modules.

Let $M$ be a free $\mathbb{Z}$-module. Let $a_1,a_2\in \mathbb{N}$ such that $\gcd(a_1,a_2)=1$. Then $${M}/{a_1a_2M}\cong {a_1M}/{a_1a_2M} \oplus {a_2M}/{a_1a_2M}.$$ Is the above statement true? ...
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2answers
37 views

A finite $\mathbb{Z}$-module whose submodules are totally ordered by inclusion.

I have the following problem: Let $M$ be a finite $\mathbb{Z}$-module such that set of the submodules is totally ordered by inclusion. Prove that there exist a prime $p$ such that $|M|=p^\alpha$ ...
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1answer
25 views

$p$ is an odd prime and $a\in\mathbb{Z}$ with $\gcd (a,p)=1$ implies $\exists x\in\mathbb{Z}:x^2\equiv a\bmod p$ iff $a^\frac{p-1}{2}\equiv 1\mod p$

Let $p$ denote an odd prime and let $a\in\mathbb{Z}$ with $\gcd(a,p)=1$. I want to show that it holds $$\exists x\in\mathbb{Z}:x^2\equiv a\text{ mod }p\;\;\;\Leftrightarrow\;\;\;a^\frac{p-1}{2}\equiv ...
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1answer
12 views

Changing the product of $V$ a $K$-space.

Let $V$ be a $K$-space with a basis $\{v_{1}, v_{2}, v_{3}, \dots\}$. Prove that there exists an product such that $x \cdot v_{i} = v_{i + 1}$ for $i \geq 1$ and $V$ get a structure of ...
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1answer
64 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?
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0answers
66 views

Divisible modules over $\mathbb Z[x]$

Divisible modules over $\mathbb Z[x]$ How do they look like? Is there any characterization similar to that over PIDs?
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2answers
28 views

map $\iota: A \rightarrow \mathbb{Q}\otimes_{\mathbb{Z}} A$, where $A$ is a finite abelian group of order n

Hi everyone I'm trying to teach myself some basic module theory using Dummit & Foote's Abstract Algebra. The second example on page 363 confuses me a bit. It says: "Let $R = \mathbb{Z}$, $S = ...
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0answers
22 views

Question about modules

Let $R$ be a ring and $R^n$ be the external sum of $n$ copies of the $R$-module $R$. Let's take $t$ elements of our $R$-module $R^n$ and call $M$ the sub-module generated by them. Then call $r$ the ...
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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 $x$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
60 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.
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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: ...
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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 ...
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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.
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1answer
26 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 ...
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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 ...
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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 ...
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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 ...
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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 $ ...
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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} & & ...
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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} ...
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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 ...
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1answer
78 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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+ ...
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2answers
74 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 $n=rs$: $$ 0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0. $$ ...
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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) + ...
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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 ...
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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 ...
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4answers
71 views

Proving that P/PJ is a projective right module over R/J

If P is a projective right module over a ring R and J is a two sided ideal of R. Prove that P/PJ is a projective right module over R/J . My idea was trying to proof that " $M$ is an $R$-module ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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1answer
42 views

Annihilator of a quotient module

Let $J$ be an ideal of $R$, and $M$ a right $R$-module. Since $Jr \subseteq J$, $M / MJ$ is naturally a right $R$-module. Since it seems relevant to another problem, I am trying to determine ...
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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?
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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 ...
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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 ...
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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?
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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 ...
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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, ...