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

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1answer
34 views

Does every free $R$-module have a maximal proper submodule?

Let $R$ be a commutative ring with $1$. We know that every finitely generated $R$-module has a maximal proper submodule. Is it true for any free $R$-module? In particular, can we do the following: ...
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1answer
27 views

Extension of an $R$-homomorphism as a sum

I want to solve this problem: Let $M$ be an injective module over a ring $R$ with identity, and $f$, $g$, $h$ are elements of the ring of $R$-endomorphisms $\operatorname{End}(M)$ such that $\ker ...
2
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2answers
40 views

Winning strategies in multidimensional tic-tac-toe

This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a ...
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3answers
62 views

Annihilator of annihilator of annihilator of a submodule

Let $R$ be a commutative ring and M be an $R$-module. The question is whether for every submodule $N$ of $M$ the equality ...
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1answer
35 views

A direct limit of pullbacks

Let an $R$-module $C$ be a direct limit of finitely presented $R$-modules $C_i$, and we have a short exact sequence as follows: $$0→A↪B\stackrel{\pi}→C→0.\qquad (S)$$ From each $C_i$ to the direct ...
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1answer
44 views

Example of a module such that every proper submodule is finitely generated but the module is not.

Let $R$ be a ring with 1 and $M$ an $R$-module. What is an example such that $M$ is infinitely generated but every proper submodule is finitely generated.
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36 views

Bass numbers of minimax modules are finite?

Let $R$ be a commutative Noetherian ring, and $M$ be a minimax $R$-module. Are the Bass numbers of $M$ are finite? (An $R$-module $M$ is called minimax, if there is a finite submodule $N$ of $M$, such ...
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1answer
69 views

Modules with finite support in $\mathrm{Max}(R)$

Let $R$ be a commutative Noetherian ring, and $M$ be an $R$-module. Is the following statement true? If $\mathrm{Supp}_R(M)$ (support of $M$) is a finite subset of $\mathrm{Max}(R)$ (the set of all ...
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1answer
38 views

Injective hull of $\mathbb{ Z}_n$ [duplicate]

What is the injective hull of $\mathbb Z_n$? I know that in case $n=p$ is prime, the injective hull would be isomorphic to $\mathbb Z_{p^∞}$, but in general case, I have no idea. Can anyone be of ...
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2answers
29 views

Cardinality of minimal generating set of a module is constant

Let $R$ be a commutative ring with unity and $M$ be a finitely presented module over $R$. Then how to show that for any minimal generating set $S$, the cardinality is same? Edit: Thanks to Martin to ...
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2answers
22 views

A direct limit concerning some homomorphisms

In an algebra text there is the following argument I am stuck in the last part of which: "Let $f:B→C$ be an epimorphism in the category of $R$-modules, and $D=∑_{n=1}^∞c_nR$ be a countably generated ...
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1answer
33 views

Existence of a maximal submodule

Let $R$ be a left Artinian ring and let $M$ be a nonzero left $R$-module. Prove that $M$ has at least one maximal submodule. I don't really know where to start. Any hint? Thanks!
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1answer
21 views

Converse to Exactness of localization of modules

It is a standard fact that if $R$ is a ring, and $$\tag{1} 0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0 $$ a short exact sequence of $R$-modules, if $S$ is a multiplicative ...
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0answers
17 views

Homomorphism of modules satisfies a monic polynomial equation

I need some help with this exercise (maybe just for the first part, or something to begin with and then I will try too solve the rest). Let $R$ be a commutative ring with identity. Let $I\subset R$ ...
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1answer
22 views

absolutely irreducible module

If L be lie algebra over F ( F is a field), I want to know what is the definition of absolutely irreducible FL-module? I have confused of several key words related to irreducible modules!? Is there ...
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1answer
33 views

Is this ring Noetherian

If $R=\{\frac m n\in{\mathbb{Q}}: \mbox{7 does not divide $n$}\}$. Is $R$ Noetherian? What i think is the ideals of $R$ are finitely generated. But how do i prove this, i have no idea. Can anyone ...
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1answer
23 views

Submodule Equality

I was wondering whether the following is true or not: Suppose $N$ and $L$ are submodules of $M$ with $N\subseteq L$. If $M/L\cong M/N$ then $L=N$. I am particularly curious about the case when ...
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1answer
25 views

Infinite dimensional Vector Spaces and Bases of Quotients

Given a basis $\mathcal{B}$ of an infinite dimensional vector space $V$, will a quotient of $V$ by a subspace $I$ necessarily have a basis? Is there a way of obtaining it using $\mathcal{B}$? My ...
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0answers
17 views

What is $M\otimes_\mathbf{Z}(N/P)\cong ?$

Let $A=\mathbf{Z}[G]$ where $G$ finite, and let $M,N,P$ be $A$-mods. Is there any particularly nice relationship for $M\otimes_\mathbf{Z}(N/P)$?
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1answer
32 views

Finding a suitable basis for a vector space

Let $k$ be a division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\mathrm{End}(V)$, defined as a ring of right operators on $V$. My questions are: (1) Why $V$ is a simple ...
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1answer
61 views

$A\otimes_RB$ is an $R$-algebra

This is Proposition 21 on page 374 of Dummit and Foote's Algebra: If $A,\ B$ are $R$-algebras, that is, they are module which have a center containing $R$, then $A\otimes_R B$ is still so where $R$ ...
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1answer
14 views

Isomorphism of a torsion product and a quotioten of torsion product.

I have the following problem: If $A'$ a submodule of the right $R$-module $A$ and $B'$ a submodule of the left $R$-module $B$, then $A/A' \otimes B/B' \cong (A\otimes B)/C$ where $C$ is the ...
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0answers
21 views

Characterization of projective modules.

Show that $R$-module $P$ is projective, iff there exist a family $\{m_{\lambda}\}_{\lambda\in\Lambda}$ of elements of $P$ and homomorphisms $\{f_{\lambda}:P\rightarrow R \}_{\lambda\in\Lambda}$ ...
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1answer
62 views

Are $\mathbb{Q}$ or $\mathbb{Z}$ flat modules?

I have the following three question about flat modules. Why is not $\mathbb{Z}$ a flat $\mathbb{Z}$-module. Why is $\mathbb{Q}$ a flat module $\mathbb{Z}$-module. I need an example of a module which ...
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1answer
29 views

Tensor product and its homomorphisms.

Given $f:A_R\rightarrow A'_R$, $g:B_R\rightarrow B'_R$ R-module homomorphism we can define $f\otimes g: A\otimes_R B\rightarrow A'\otimes_R B'$ such that $(f\otimes g)(a\otimes b)=f(a)\otimes g(b)$ ...
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1answer
23 views

Help understanding statement relating to structure of modules over PIDs

Lemma IV.6.11 of Hungerford's Algebra is Lemma 6.11. Let $R$ be a principal ideal domain. If $r \in R$ factors as $r = p_1^{n_1} \cdots p_k^{n_k}$ with $p_1,\ldots,p_k \in R$ distinct primes and ...
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0answers
27 views

Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?

so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...
2
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0answers
67 views

If $R$ is a domain and $M$ a finitely generated $R$-module, is it true that $\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M$?

Let $R$ be a domain and $M$ a finitely generated $R$-module. Let $M^{*}=\hom_{R}(M,R)$. Let Tor$M$ be the torsion submodule of $M$. It it true that $$\displaystyle\bigcap_{f\in ...
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1answer
34 views

$0\to L\to M\to N\to 0 $ is split if $0\to {\rm Hom}_R(D,L)\to {\rm Hom}_R(D,M)\to {\rm Hom}_R(D,N)\to 0$ is exact for any $D$.

Prove that $0\rightarrow L\rightarrow M\stackrel{\phi}\rightarrow N\rightarrow 0 $ is split if $$0\rightarrow {\rm Hom}_R(D,L)\rightarrow {\rm Hom}_R(D,M)\rightarrow {\rm Hom}_R(D,N)\rightarrow 0$$ ...
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0answers
42 views

Polynomial Ideals and Transverse modules

I have a given ideal and I want to find the "smallest" ideal so that when I add it to the original one, I get another certain ideal. Let $\mathcal{E}$ be the ring of smooth function germs ...
2
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2answers
58 views

Consider $\mathbf{Z}G$, $G$ finite. If the characters of two $\mathbf{Z}G$-modules are equal, does it follow that the modules are isomorphic?

So I have recently started to delve into integral representation theory and I was wondering if a particularly useful theorem survives the transition to integral rep theory. Basically, suppose we have ...
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2answers
56 views

Existence of module homomorphism $\phi : M \rightarrow N$ such that $\Phi =\phi^2$

This is a homework problem and I have solved part (a), but I am not sure how to approach part (b). Should I approach it the way how the universal property of free modules are defined? Any hint(s) ...
4
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1answer
121 views

Free modules basic understanding problem

I have been told that the $\mathbb{Z}$ module $\mathbb{Z}/2\mathbb{Z}=\{0,1\}$ isn't free. For a module to be free, there must exist a subset such that every element is expressible as a finite linear ...
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1answer
16 views

Irreducibles - difficulty with the definition

I'm working from the definition that in an integral domain $R$, an irreducible is an element $p$ such that if $p=xy$ then either $x$ or $y$ is a unit. In certain proofs on my course, the lecturer has ...
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1answer
28 views

Unqiue Factorization Domains, is the product finite?

Having looked around a bit, the most common definition of a UFD is an integral domain such that any element can be expressed as a product of a unit and irreducible elements, and that this ...
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1answer
60 views

How does the free abelian group of $M \times N $ have $M\times N$ as basis in construction of tensor product of modules?

With M and N being R-modules, how does Z(M,N) have $M\times N$as a basis and therefore becomes a free abelian group? Consider the element n(m,0) for an element$\,m\in M $ of order n. This is zero ...
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1answer
51 views

Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
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0answers
50 views

Direct proof that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p^2\mathbb{Z}$-module.

I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following. We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} ...
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1answer
19 views

Finitely generated quotient module isomorphism

In an assignment, I was asked to prove that if $M$ is an $R$ module and $N\subseteq M$ is a submodule such that $M/N$ is finitely generated and free, then there exists a submodule $F$ of $M$ which is ...
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0answers
25 views

Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
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1answer
21 views

Inclusion induce an isomorphism

I have a small question If $E,F$ are two sub-modules of a module $G$, how to prove that the inclusion $E \rightarrow E+F$ induces an isomorphism $(E/E \cap F) \simeq (E + F) / F$ Please Thank you ...
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0answers
45 views

Structure theorem of modules in the graded case

I ran into an exercise in D. Passman's book A course in ring theory (page 130) asking me to prove the structure theorem of modules over PIDs in the graded case. Could anyone point me in the right ...
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1answer
17 views

Finitely generated quotient module

In an assignment question, I'm a bit stumped by the following: Let $R$ be a commutative ring, $M$ be an $R$-module and $N$ an $R$-submodule of $M$. I was first asked to prove that if both $N$ and ...
2
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3answers
61 views

Contrasting definitions of bimodules? An illusion?

Recently my definition of a bimodule over a $k$-algebra has been challenged and I believe both definitions to be equivalent, am I wrong? Notation: $k$ is a commutative ring and $A$ is a (unital ...
2
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1answer
128 views

Is there vector space $\mathbb R $ over $\mathbb R $ with dimension $m\neq1$?

I am searching a new scalar product that with this scalar product the vector space $\mathbb R $over $\mathbb R $ with ordinary vector sum has dimension $m$ , which $m$ is any arbitrary natural ...
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1answer
67 views

Any module is the colimit of its finitely generated submodules.

Fact A : If $E_i$ is a directed system of $R$-modules, and $F$ is another $R$-module. Then $$\varinjlim(F\otimes E_{i})=F\otimes(\varinjlim E_{i}).$$ Fact B : "Every module is a direct limit of its ...
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1answer
38 views

Question about Universal Mapping Property

Now I seem to understand the construction of the tensor product of $R$-Modules by defining $$F_R(M \times N) := \bigoplus_{(m,n) \in M \times N} R\delta_{(m,n)}$$ and constructing the submodule $D$ in ...
2
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3answers
125 views

Isomorphic quotient of a module over Noetherian commutative ring

I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...
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0answers
14 views

$f_I(R)$ and $ f_I(M)$

The question is special case of this question. So, for background, see it. Is there a relation between $f_I(R)$ and $ f_I(M)$ ? Any hint, reference will be helpful. Thanks.
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0answers
56 views

Tensor product of free modules over free algebra

Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$. Question: Under what ...