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

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1answer
71 views

Are torsion modules always finite?

I was just wondering...are all torsion modules finite? Because, for instance, if the annihilator is (a), where a is nonzero, then all multiples of a are zeros...so that's like a cycle, right? Thanks ...
2
votes
1answer
63 views

Big test coming up, cant figure this one out: isomorphism between free R-modules

Let $R$ be a commutative ring with a unit. prove that $\operatorname{Hom_R}(R^n,R^m)$ is a free $R$-module of rank $n*m$, please help :] I thought I'd go with $\operatorname{Hom_R}(R^n,R^m)$ $\cong$ ...
0
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2answers
393 views

Free Modules…

From my textbook: A free $R$-module is "A left $R$-module $F$ is called a free left $R$-module if $F$ is isomorphic to a direct sum of copies of $R$..." I know that another definition of a free ...
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0answers
251 views

ISBN Code - Construction, check digit example

I am doing a homework example and would like to ask your opinion on the correctness of the example: How is the ISBN 3-519-42227-1 constructed, at what point is the check digit and how to calculate ...
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0answers
74 views

$x \otimes S \cong y \otimes S$

Let $R$ be a commutative ring and $M$ a $R$-module. Which conditions could one find for elements $x,y \in M$, so that the isomorphism $$ x \otimes S \cong y \otimes S $$ as $R$-modules is true? $S$ ...
1
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1answer
56 views

About the basis of a module

Let $n\in \mathbb Z, n\ne 0$. Prove that $ \mathbb Z/n\mathbb Z$-module $\mathbb Z/n\mathbb Z$ has a basis, but $\mathbb Z$-module $\mathbb Z/n\mathbb Z$ hasn't any basis. Hope everyone help me with ...
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2answers
76 views

Show by example that this need not be true if we do not assume that the groups are finitely generated

Let $G, H,$ and $K$ be finitely generated abelian groups. If $G \times K \cong H \times K$, show that $G \cong H$. Show by example that this need not be true if we do not assume that the groups ...
3
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3answers
323 views

Find two short exact sequences of abelian groups such that two of them are isomorphic, however the third is not

I'm just trying to solve an exercise from "A course in homological algebra" by Hilton and Stammbach, however I couldn't find any example. Find two short exact sequences of abelian groups $0 ...
3
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0answers
77 views

Free submodule of the intersection of free direct summands.

I have slightly more assumptions than the "standard" question (see e.g. this one) so the usual counterexamples that come to mind don't work (I believe). Let $R$ be a ring and $M$ a free $R$-module. ...
2
votes
1answer
113 views

Dual notion of free module

I was just introduced to the notion of projective module (through Matsumura, Commutative ring theory), and there it is said that the reason that there is no easy description for injective modules ...
8
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4answers
268 views

The connection between decomposition of $\ker(f_\phi g_\phi)$ and the Chinese remainder theorem

Original problem Suppose $K$ is a field, $f,g\in K[x]$ and $\gcd(f,g)=1$. $V$ is a linear space based on $K$ and $\phi\in\operatorname{End}(V)$. Notation $f_\phi$ and $g_\phi$ denote linear ...
0
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1answer
182 views

Ex. of a finitely generated module without a finite basis.

I am working my way through Linear Algebra - Hoffman and Kunze. There is a very brief introduction to Modules in the chapter on Determinants. The authors state "..a module may be finitely generated ...
3
votes
1answer
120 views

Natural homomorphism $\mathfrak{a} \prod E_\lambda \to \prod \mathfrak{a} E_\lambda$

Let $A$ be a ring (we don't assume that $A$ is commutative), and $\frak a$ a left ideal of $A$. Let $(E_\lambda)$ be a system of left $A$-modules. There is a natural homomorphism of modules $$ \phi ...
7
votes
3answers
487 views

If $M$ is an artinian module and $f$ : $M$ $\mapsto$ $M$ is an injective homomorphism, then $f$ is surjective

If $M$ is an artinian module and $f$ : $M$ $\mapsto$ $M$ is an injective homomorphism, then $f$ is surjective. I somehow found out that if we consider the module $\mathbb Z_{p^{\infty}}$ denoting ...
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2answers
79 views

Questions related to exterior powers

I have two problems related to exterior powers, and I have little background on the exterior algebra of an $R-$mod. Problem 1. For any ideal $J_i\subset R$, which is a commutative ring, consider ...
5
votes
3answers
418 views

(geometric/intuitive) interpretation of ext

In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, e.g. the connection between n-th extensions and ext. But I don't have a feeling about ...
6
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2answers
190 views

Why is this statement about generators of groups true?

Let $G$ be free abelian of rank $n$ and $H \subseteq G$ a subgroup also of rank $n$. It is known that $G/H$ is finite, in fact a direct sum of at most $n$ cyclic groups. Thus we can write $$G/H = ...
0
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2answers
111 views

Suppose R is a commutative ring, $A \in R_n$, and the homomorphism $f: R^n \to R^n$ defined by $f(b) = Ab$ is surjective. Show $f$ is an isomorphism.

How would I go about showing this? Suppose $R$ is a commutative ring, $A \in R_n$, and the homomorphism $f: R^n \to R^n$ defined by $f(b) = Ab$ is surjective. Show $f$ is an isomorphism. (Edit: ...
3
votes
1answer
468 views

Support of a module and submodule

$\newcommand{\Supp}{\operatorname{Supp}}$ $\newcommand{\Spec}{\operatorname{Spec}}$ $\newcommand{\Ann}{\operatorname{Ann}}$ Let $A$ be a commutative ring with 1. Let $M$ be a finitely-generated ...
19
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3answers
363 views

Bound on nilpotency index of endomorphisms

Let $A$ be a Noetherian ring (commutative with $1$) and $M$ a finitely generated $A$-module. I want to show that there exists a bound $n$ such that for every nilpotent endomorphism $T : M \to M$ we ...
5
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3answers
374 views

Expressing, in terms of $I$ and $M$, the $R$-modules $\mathrm{Hom}_R(R/I,M)$, $\mathrm{Hom}_R(M,R/I)$, $\mathrm{Hom}_R(I,M)$, $\mathrm{Hom}_R(M,I)$

Let $R$ be a commutative unital ring, $I$ an ideal of $R$, and $M$ a $R$-module. It is known that $R/I \otimes_R M \cong M/IM$. Also, $\mathrm{Hom}_R(R,M)\cong M$. Is there some similar formula for ...
0
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3answers
81 views

$M=M_1\oplus M_2$. Then show that $M/M_1$ is isomorphic to $M_2$ and $M/M_2$ is isomorphic to $M_1$

Show that if $M$ is a direct sum of $M_1$ and $M_2$ then $M/M_1$ is isomorphic to $M_2$ and $M/M_2$ is isomorphic to $M_1$.
1
vote
1answer
57 views

Let $N$, $K$ be sub-modules of $M$ with $I=\mathrm{Ann}(N)$, $J=\mathrm{Ann}(K)$. Show $I+J$ is a proper subset of $\mathrm{Ann}(N \cap K)$.

Let $N$ and $K$ be sub-modules of $M$ with $I=\operatorname{Ann}(N)$ and $J=\operatorname{Ann}(K)$. Show that $I+J$ is a proper subset of $\operatorname{Ann}(N \cap K)$.
2
votes
1answer
86 views

What do you call this generalization of a module?

A module $M$ is an Abelian group with the extra property that any element $m \in M$ can be multiplied by any element $r$ in a ring $R$. I want to relax this definition so that the $M$ need not be an ...
2
votes
1answer
157 views

Why if the maximal ideal is nilpotent then this module is free?

On matsumura is proved the following proposition: let $(R,m)$ be a local ring and $M$ a flat $R$-module. If $x_1,\ldots,x_n\in M$ are such that their images $\bar{x}_1,\ldots,\bar{x}_n$ in ...
6
votes
1answer
249 views

The support of a non finitely generated module

Let $R$ be a ring and $M$ an $R$-module. If we suppose $M$ finitely generated then (following Matsumura) if we write $M=Rm_1+\cdots+Rm_n$ we have: $p\in\mathrm{Supp}\;M$ if and only if $M_p\neq0$ if ...
4
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1answer
120 views

Defining the image of a map

Let $\mathcal{A}$ be an abelian category. For a morphism $f:A\rightarrow B$, its image is usually defined in the following way: a map $i:I\rightarrow B$ is called an image of $f$ if it is a kernel of ...
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2answers
75 views

Why is $rM$ a submodule?

Let $R$ be a ring and $M$ be a $R$-module. In Example 5.10 of Algebra: Chapter 0 that for $r\in R$, \begin{equation} rM=\{r\cdot m:m\in M\} \end{equation} is a submodule of $M$. I have no problem ...
4
votes
1answer
86 views

If Hom(M,-) is stable under base change, is then M f.g. projective?

Let $R$ be a commutative ring. Let $M$ be an $R$-module with the following property: For every commutative $R$-algebra $A$ and every $R$-module $N$ the canonical map $\mathrm{Hom}_R(M,N) \otimes_R A ...
4
votes
1answer
457 views

Additive functor over a short split exact sequence.

$0\to A'\stackrel{f}{\longrightarrow} A \stackrel{g}{\longrightarrow} A''\to 0$ is a short split exact sequence, where $A'$, $A$, $A''$ are $R$-modules, and $T$ is an additive functor from ...
-2
votes
1answer
276 views

Example of torsion-free module

Let $B=k[X,Y]/(Y)$ be a module over $A=k[X,Y]/(XY)$ (under the natural ring-homomorphism). Give an example of torsion-free module $M$ over $A$ such that $M \otimes_{A} B$ is not torsion free over $B$. ...
1
vote
1answer
167 views

Every subgroup of finite rank of $\mathbb Z^\mathbb N$ is free abelian

I want to prove that every subgroup of finite rank of $\mathbb Z^\mathbb N$ is free abelian. A group $G$ has finite rank $N$ if there exists a subset $ \{e_1,..,e_N\}$ of $G$ such that: $(1)$ If $a_i ...
1
vote
1answer
66 views

proving that a action of hopf algebra k(G) on A implies a G-grading on A

Let $k(G)$ be the hopf algebra of functions on $G$ with values in $k$ with pointwise multiplication and a comultiplication given by $\Delta(f)(x,y) = f(xy)$ and let $A$ be a $k$-algebra. I read (in ...
1
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1answer
46 views

Cancellation law of morphisms?

The title is probably misleading, but I wasn't quite sure how to boil my question down to one line. My problem is this: Assume I have two projective $R$-modules, $P$ and $Q$, and epimorphisms ...
4
votes
0answers
42 views

Normal Form problem (in a module over a PID)

Let $A,B$ be $n\times n$ matrices with entires in a PID $D$ and $\det AB\neq 0$. Suppose diag$\{a_i\}$, diag$\{b_i\}$, and diag$\{c_i\}$ are normal froms for $A$, $B$, and $AB$. In particular, ...
2
votes
1answer
84 views

is the idele class group a flat Z-module?

If $K$ is a local or global field and I define $C_K$ to be the idele class group if $K$ is global and I let $C_K=K^{\times}$ is $K$ is local, then as a $\mathbb{Z}$-module will it be flat? i.e. If $$ ...
2
votes
2answers
128 views

Why is every ideal in $\mathbb Z[ i ] / (5)$ principal?

Let $R = \mathbb Z[ i ] / (5)$ . Prove that any ideal in $R$ is principal. Any ideas on how to prove this?
2
votes
1answer
49 views

$x^2 +1$ is invertible in $Q[x]/(x^4 −3x^2 +6x)$

Why is $x^2 +1$ is invertible in $Q[x]/(x^4 −3x^2 +6x)$? Any ideas on how to prove this?
2
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2answers
143 views

The classification theorem for modules over $\mathbb{Z}[i]/(5)$

Let $R = \mathbb{Z}[i]/(5)$. It is obvious that $R$ is not an integral domain, and any ideal in $R$ is principal. Now I want to prove the following classification theorem for modules over $R$ : ...
1
vote
1answer
123 views

Property of an operator in a finite-dimensional vector space $V$ over $R$

Let $L: V\to V$ be an operator in a finite-dimensional vector space $V$ over $R$. For any $n \geq 0$, let $K_n = \ker (L^n)$, $I_n = \mathrm{Im}(L^n)$. (a) Prove that there exists $N$ such that ...
0
votes
3answers
350 views

$R = \mathbb{Z}[ i ] / (5)$ is not an integral domain? Why?

Let $R = \mathbb{Z}[ i ] / (5)$ . How should I prove that $5 = (2+i) (2-i)$ is a prime factorization in $\mathbb{Z}[i]$? Can we deduce from this that R is not an integral domain? How? I know that ...
4
votes
4answers
652 views

Noetherian module implies Noetherian ring?

I know that a finitely generated $R$-module $M$ over a Noetherian ring $R$ is Noetherian. I wonder about the converse? I believe it to be false and I am looking for counterexamples. Also I wonder if ...
2
votes
1answer
88 views

Same left KG-module induced by nonisomorphic G-sets

For any field $K$, a finite group $G$, and left $G$-set $X$, $KX$ has a obvious left $KG-$ module structure, where $KG$ is the usual group algebra generated by the group $G$. The question is Find ...
1
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1answer
128 views

Is $R = Q[x] / (x^4 - 3x^2+ 6x)$ isomorphic to a direct sum of two fields?

Let $R = Q[x] / (x^4 - 3x^2+ 6x)$. How can we prove that $x^2 + 1$ is invertible in $R$? How can we prove that $R$ is isomorphic to a direct sum of two fields?
0
votes
1answer
70 views

Question about Vector sapces [duplicate]

Possible Duplicate: Property of an operator in a finite-dimensional vector space $V$ over $R$ How do we prove the stabilization property:there exists $N$ such that for all $n \geq N$, we ...
1
vote
1answer
137 views

Find the rank of Hom(G,Z)?

(1) Prove that for any finitely generated abelian group G, the set Hom(G, Z) is a free Abelian group of finite rank. (2) Find the rank of Hom(G,Z) if the group G is generated by three generators x, ...
1
vote
1answer
156 views

Submodule of a finitely generated module over an artinian ring

I have a left artinian ring $R$ and a finitely generated left $R$-module $M$, and a submodule $A$ of $M$. My question is : is $A$ necesseraly finitely generated ? (and is there a direct proof of this ...
2
votes
1answer
167 views

Non-Free Finitely Generated Injective Modules over a Local Ring

I was wondering if someone could be so kind as to provide an example of a local ring $ (R,\frak{m}) $ and a non-free finitely generated injective module over $ R $. Thank you very much! I tried ...
4
votes
1answer
219 views

How does one show that this tensor product is not torsion-free?

I am having trouble showing that a particular tensor product is not torsion-free. Let $ R = k[[x,y]] $, where $ k $ is a field (this is the ring of formal power series in $ x $ and $ y $ with ...
10
votes
0answers
225 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we ...