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

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1answer
69 views

The ideal $I=(3,2+\sqrt {-5})$ is a projective module

Let $R=\mathbb Z[\sqrt{-5}]$ and $I=(3,2+\sqrt {-5})$ be the ideal generated by $3$ and $2+\sqrt{-5}$. I'm trying to prove that $I$ is a projective $R$-module. I'm using the lifting property ...
0
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1answer
49 views

If M is free with a finite basis then every basis of M over R is finite and has the same number of elements.

Stuck on a proof in my lecture notes. Proposition: Let $R$ be a commutative ring and let $M$ be an $R$-module. If $M$ is free with a finite basis then every basis of $M$ over $R$ is finite and has ...
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1answer
34 views

$f:M\rightarrow N$ module homomorphism, $(N/\mathrm{Im}f)_m=N_m/\mathrm{Im}f_m$

$f:M\rightarrow N$ is an $R$-module homomorphism and $f_\mathfrak{m}:M_\mathfrak{m}\rightarrow N_\mathfrak{m}$ is the induced $R$-module homomorphism $$f_\mathfrak{m}(m/s)=f(m)/s$$ where ...
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1answer
41 views

Dimension of dual module

Let $R$ be a finite dimensional semisimple $k$ algebra ($R$ is not necessarily commutative) and $M$ be an $R$-bimodule such that $M$ has finite dimension over $k$. Define ...
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1answer
25 views

Localization of modules; sums commute

Let $B_i$ be a family of $R$ submodules of a module $B$. Show $$S^{-1}(\sum_{i\in I}B_i) = \sum_{i\in I} S^{-1}B_i$$ as $S^{-1}R$ submodules of $S^{-1}B$. If it helps, we know that ...
2
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2answers
14 views

Quotienting a direct sum by one of its factors

If we have a direct sum of $R$-modules. say $M_1\oplus M_2$ does it then follow that $(M_1\oplus M_2)/M_1\cong M_2$ This seems like it should be the case but I can't think of a way to prove ...
2
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2answers
34 views

How to prove that $M \cong \mathrm{ker}(f) \oplus A$?

Let $M$ be an $A$-module, $A$ commutative ring. $ f : M \to A$ is a surjective morphism, then prove that $$M \cong ker(f) \oplus A$$ Any hint ?
2
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1answer
38 views

How can I prove this is an $R$-module?

Let $R$ be a ring, $r_1\ldots,r_n\in R$ and $P(r_1,\ldots,r_n)=\{(c_1,\ldots,c_n)\in R^n|\sum c_ir_i=1\}$. I'm trying to prove that $P(r_1,\ldots,r_n)$ is an $R$-module, how would be a good candidate ...
3
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1answer
43 views

The number of submodules of $M\oplus M$

Let $M$ is a simple $R$-module. Then what we can say about the number of submodules of $M\oplus M$? it can be infinite 2 3 4 I say that if $N$ be a submodule of $M\oplus M$, then for projective ...
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1answer
54 views

Show that a finitely generated module is trivial if it's equal to a maximal ideal times itself

Let $R$ be any commutative ring which is local. $M$ a finitely generated $R$ module, and $ I \subset R$ a maximal ideal. How to show that $ M = IM$ $ \Longrightarrow$ $M = 0$ in an elementary way? And ...
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1answer
27 views

Why we cannot in general define the product of two submodules

why we can define product of two ideals of a commutative ring,but we can't in general define the product of two submodules? ...
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1answer
21 views

Show $M(\mathfrak{p})$ is an $R$-submodule of $M$

Let $R$ be a PID, $\mathfrak{p}$ be a prime ideal in $R$, and $M$ an $R$-module. Then $$M(\mathfrak{p})=\{m\in M \mid \operatorname{Ann}_Rm=\mathfrak{p}^j, \, \text{ for some } j\geq 0 \}$$ is ...
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1answer
33 views

How to prove M/Ker(f) & Im(f) are isomomorphic

f:M->N us an A-module homomorphism. A is a commutative ring. How to prove M/Ker(f) and Im(f) are isomomorphic I can't prove this statement. But if A is a field,it's not hard to be proved. For ...
5
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1answer
63 views

Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module?

I'm confused. Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module? We know that $\Bbb Z_{p^{\infty}} \subset \Bbb Q/\Bbb Z$ is artinian. The following argument is true or not ? $\mathbb Q / \mathbb Z$ ...
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0answers
31 views

Comodules as a functor category

Let C be a comonoid in some preadditive monoidal category $\mathfrak{C}$, then how can we express the category of C-comodules, in terms of some sort of functor category? I mean is there a similar ...
4
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1answer
117 views

How to calculate $Ext(M,N)$?

I am confused about the calculation of $\text{Ext}(M,N)$. If $N$ is a fixed module and if we consider the projective resolution $$\cdots \to C_1 \to C_0 \to M \to 0,$$ then $\text{Ext}_n(M,N)$ is the ...
2
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1answer
32 views

Looking for a concise treatise on the $\mathbb Z^n$ as a $\mathbb Z$-module

I am working on something which turned out to be isomorphic to the $\mathbb Z$-module $\mathbb Z^n$ and want to efficiently learn about its properties without needing to deal too much with more ...
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0answers
46 views

Hom functor and localization

I saw in some book the following question Let $R$ be a commutative ring and $P$ a prime ideal of $R$. Prove that $$\mathrm{Hom}_R(M,N)_P \simeq \mathrm{Hom}_{R_P}(M_P,N_P)$$ if $M$ is finitely ...
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2answers
123 views

Relations between monoids and modules?

What is the relation between monoids and modules? Are they completely different algebraic structures, or is there a kind of inclusion relation like "elements of a module are also elements of a ...
2
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1answer
40 views

Action of the center of an enriched category on a module as an enriched functor

If $M$ is a left-module over a ring $R$, then $M$ carries an action of $Z(R)$, the center of $R$, since it is a subring of $R$, by restriction of scalars. Since $Z(R)$ is commutative we may view this ...
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1answer
125 views

Classify Artinian $\Bbb Z$-modules

How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful. I have seen the ...
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1answer
74 views

Showing $M\cong M'\oplus M''$ given an exact sequence

I am struggling with the following question: $R$ is a ring. $$M'\overset{f}{\longrightarrow} M\overset{g}{\longrightarrow} M''$$ are homomorphisms of $R$-modules such that for any $R$-module $N$, the ...
2
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1answer
29 views

When are maps between Hom sets induced?

I'm trying to better understand $R$-module homomorphisms, and I know that say, an $\, f:M\to N$ induces $\, f_*:Hom_R(V,M)\to Hom_R(V,N)$ or $\, f^*:Hom_R(N,V)\to Hom_R(M,V)$. What I'm wondering is, ...
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0answers
26 views

Relationship between two full lattices over a Dedekind domain

This question is from page 84 of Curtis and Reiner's book 'Methods of Representation Theory' Vol.1. Let $R$ be a Dedekind domain with quotient field $K$. Let $M$ and $N$ be finitely generated ...
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0answers
28 views

What is the most generic algebraic structure for which we can define a tensor product? [duplicate]

We can define a tensor product of two vector spaces. But vector spaces are themselves modules and we can also define a tensor product of two modules. My question is the following: are modules the ...
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0answers
87 views

Almost continuous and almost cocontinuous functors

I would like to consider Hom-like functors $H:\mathcal{A}\rightarrow\mathcal{B}$ defined between abelian categories or, more specifically, between module categories, in the following sense: $H$ is ...
5
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1answer
103 views

Artin exercise on free modules homomorphism

2.3 Let $A$ be the matrix of a homomorphism $\varphi:\mathbb Z^n\to\mathbb Z^m$ of free $\mathbb Z$-modules. (a) Prove that $\varphi$ is injective if and only if the rank of $A$, as a real ...
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67 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
2
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1answer
110 views

Annihilators and exact sequences

Let $R$ be a commutative ring. Let $M_1$, $M_2$ and $M_3$ be $R$-modules. Let the following sequence be exact: $$0\longrightarrow M_1 ...
2
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1answer
53 views

Coproduct diagram for tensor product

Is it true that in the category $R$-Mod of $R$-modules (and $R$-module homomorphisms), the diagram $$ M \longrightarrow M \otimes_{R} N \longleftarrow N,$$ where the arrows are the maps $m \to m ...
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1answer
103 views

An ideal in a domain is free if and only if it is principal [closed]

Let $R$ be a domain. Show that every ideal in $R$ is $R$-torsion-free, but is free if and only if it is principal. In particular, show that $R$ is a PID if every submodule of a free module is free. ...
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1answer
37 views

Surjective endomorphism of an $R$-module is injective.

I know this is a duplicate question. However, I haven't seen anything that invokes the isomorphism theorem. Here's my idea: By the isomorphism theorem we have that $M/\ker\varphi \cong ...
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0answers
47 views

Global dimension of endomorphism rings

Does anybody have an idea on how the global dimension of the endomorphism ring of a module over a (nice enough) ring is related to the global dimension of the endomorphism ring of its projective ...
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2answers
55 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...
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0answers
35 views

book suggestion on module theory

I want to start module theory from the very beginning. I have a little bit of idea about what module is but I haven't really solved a handful of problems. Could someone please suggest me a good ...
3
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1answer
96 views

Modules and submodules

Let $G=S_n = Sym_n$ be the symmetric group and $V$ a vector space with basis $\{v_1,...,v_n\}$, then $V$ is a module with action defined by $g$. $v_i$=$v_{g(i)}$ for 1$\leq$i $\leq$ n and extending ...
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1answer
37 views

If an $A$-module $M$ is locally finitely presented (resp. related) then $M$ is finitely presented (resp. related)

In this question I want to ask for a better proof than the one I am about to give for the statement with finitely presented, and inquiry if the statement is also true for the notion of finitely ...
2
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1answer
19 views

Transferring Exactness

If $$\begin{matrix}0&\rightarrow&A&\rightarrow&B&\rightarrow&C&\rightarrow&0\\ &&\downarrow&&\downarrow&&\downarrow\\ ...
3
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1answer
52 views

Determinant of long exact sequence

Let the following be a long exact sequence of free $A$-modules of finite rank: $$0\to F_1\to F_2\to F_3\to...\to F_n\to0$$ I want to show that $\otimes_{i=1}^n (\det F_i)^{-1^{i}} \cong A$, where ...
4
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1answer
52 views

Left adjoint to direct sum?

In the category of vector spaces, is there some endofunctor $F$ satisfying $$\mathrm{Hom}_k(M,\underset{i \in I}{\bigoplus} k) \cong \mathrm{Hom}_k(F(M),k)$$ for every $k$-vector space $M$?
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1answer
45 views

Is there always an epimorphism from a free $A$-module to a given one?

Let $A$ be a commutative ring. Given an $A$-module $P$, can I always construct a free module $M$ and a surjective homomorphism $f:M\rightarrow P$? I was thinking something along the lines of tensor ...
2
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2answers
35 views

Direct product commutes with coproducts?

Do direct products commute with the direct sums of vector spaces? Basically is $\underset{i \in I}{\prod} \underset{j \in J}{\bigoplus}M_{i,j} \cong \underset{j \in J}{\bigoplus}\underset{i \in ...
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1answer
42 views

Coproducts and direct products

Is the arbitrary direct sum of modules a submodule of their coproduct? Ie is $\underset{i \in I}{\coprod} M_i \cong \underset{i \in I}{\bigoplus} M_i$... if not then if each $M_i$ where to be ...
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1answer
30 views

direct products and direct sums of skew fields

If F is a skew field then are the arbitrary direct sums of F-modules isomorphic to their direct products (over the same index). I mean, if R is a division ring, and $\{M_i\}_{i\in I}$ some familly ...
3
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1answer
54 views

“Vector spaces” over a skew-field are free?

Are modules over a skew field free? That is, if $F$ is a skewfield then can any module $M$ be written as $\underset{i \in I}{\bigoplus} F$ for some indexing set $I$?
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1answer
69 views

The cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$

I want to calculate the cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$, $k,n \in \mathbb{N}$. When $n$ is prime, it is easy to calculate, so I want to know the general case.
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1answer
21 views

pushforward of affines

Suppose $A\rightarrow B$ is a morphism of rings (commutative with unit, not necc. Noetherian) and $f: \rm{Spec}\; B\rightarrow \rm{Spec} \;A$ the canonical morphism of schemes. Is it true that ...
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1answer
42 views

Tensor of quotients question

Let $R = \mathbb{Z}[x]$ be the 1-variable polynomial ring over the the integers, $q$ and $p$ arbitrary prime numbers, $R_1 = \mathbb{Z}[[x]]/(p)$, and $R_2 = \mathbb{Z}[x]/(q, x)$. Question: what ...
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1answer
27 views

Cotensor and counit?

If M is a C-bicomodule, then considering C as a $C$-bicomodule also, is $M \square_C C \cong C$, where $\square_C$ is the cotensor product in $^C\mathscr{M}^C$.
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0answers
15 views

Commutativity of comodules

If C is a cocommutative R-coalgebra, R is some commutative semi-simple artinian ring and A and B are C-bicomodules, then is $A\otimes_R B \cong B \otimes_R A$ as $R-modules$. However, this also true ...