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

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1answer
34 views

When does a real inverrtible matrix send a lattice in $\mathbb{R}^2$ to itself?

There is a question in Artin asking when does an inverrtible 2x2 real matrix send a lattice to itself. The issue I have is with this question not being specific as to what kind of answer is ...
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0answers
42 views

When do three complex numbers define a lattice?

Given three complex numbers, when is the set of all integer linear combinations of them a lattice? Since lattices can be defined by only two basis elements, I suppose the answer must be that one ...
2
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1answer
51 views

Subring of a commutative Noetherian ring

We know that it's possible subring of the commutative Noetherian ring become not Noetherian (for example: Subring of a Noetherian ring need not be Noetherian?). But if $S$ be a subring of ...
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1answer
97 views

Is this module injective? [duplicate]

Here's the problem: (This problem is from Hungerford's Algebra Chapter 5 exercise 6.7 ) Let R be a principal ideal domain and $p$ a prime in $R$ and $n$ a positive integer. Then $R/(p^{n})$ is an ...
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1answer
24 views

Module isomorphism from $R$ to $R \oplus R$ for a certain ring $R$

My textbook says: Let $R$ denote the set of infinite–by–infinite, row– and column–finite matrices with complex entries. Show that $R \cong R \oplus R$ as $R$–modules. So for $A, B \in R$, I tried ...
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2answers
50 views

What are the semisimple $\mathbb{Z}$-modules?

What are the semisimple $\mathbb{Z}$-modules? Comments: I think they are direct sums of copies of such $\mathbb{Z}_p$'s, where $p$ is a prime number. I believe it is, but I can not prove.
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1answer
32 views

generating system of the kernel of a module-transformation

Let $G ≠ 1$ be a group and A a commutative ring. Now, the group ring $A[G]$ is naturally an A-module. Next, let's consider the transformation: $$\phi: A[G] \to A, \sum_{g \in G} a_gg \mapsto \sum_{g ...
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1answer
36 views

Show that if $M$ is a semisimple artinian module then $M$ is finitely generated.

The exercise is as follows: Show that for a semisimple module $M$ over any ring, the following conditions are equivalent: $(1)$ $M$ is finitely generated; $(2)$ $M$ is Noetherian; $(3)$ $M$ is ...
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1answer
60 views

If R is a Noetherian ring, is it true that there exists an integer N such that every ideal of R is generated by at most N elements? [duplicate]

I think, as R is an ideal of itself, and R is Noetherian thus R is finite generated, assume it is generated by N elements, then every ideal is generated by at most N elements. But my conclusion is ...
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1answer
22 views

Constructing a module homomorphism

Let $\phi: M \to M'$ be a homomorphism of left R-modules, and let $N' \subset M'$ be a submodule. Construct a natural module homomorphism: $\tilde{\phi}: M/\phi^{-1}(N') \to M'/N'$ and show that ...
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2answers
23 views

$\displaystyle \mathbb KG\simeq \frac{\mathbb K[t]}{\langle t^n-1\rangle}$ as algebras?

Suppose $G$ is a cyclic group of order $n<\infty$. How can I show the isomorphism of algebras: $$\mathbb KG\simeq \frac{\mathbb K[t]}{\langle t^n-1\rangle}?$$ Above $\mathbb K$ is a field, ...
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0answers
41 views

Colimits in $Ch_R$, help with a step of the proof

I want to prove that the category of chain complexes of R-modules admits small colimits. I was told to try proving that the chain complex defined degree wise as the colimit of the modules of the same ...
2
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1answer
29 views

Number rings as free module over base ring

Let $K \subset L$ be number fields and $\mathcal{O_K}, \mathcal{O_L}$ the corresponding rings of algebraic integers. Further let dimension$(L/K)$ = n as a vector space. If $K$ is a PID, then ...
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1answer
27 views

Finite tensor product of infinite $R$-modules

I was challenged to find a ring $R$ and infinite $R$-modules $A$ and $B$ such that $A \otimes_R B$ is a finite but nontrivial abelian group. I can think of many examples that give $0$, but I'm ...
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1answer
25 views

Regard naturally as modules.

Suppose that $I$ is a two-sided ideal in the ring $R$, and that $M$ is a module over the quotient ring $R/I$. Why can we naturally regard $M$ as a $R$-module that is annihilated by $I$? Conversely, ...
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1answer
59 views

(Hopefully) Simple question about the exterior algebra functor

I have some (hopefully super) basic questions about the exterior algebra functor $$ \wedge:R\text{-Mod}\rightarrow R\text{-Alg}. $$ As I (think I) understand it, if one considers it as a functor ...
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1answer
16 views

Is the following short exact sequence always split?

Is it always true that the following short exact sequence of modules split? $0\to N\to M\to M/N\to0$
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1answer
46 views

Does the bicategory of bimodules really have left Kan extensions?

Let $\mathsf{Bimod}$ denote the bicategory whose $0$-cells are rings $A$, a $1$-cell $A \longrightarrow B$ is an $(A,B)$-bimodule and a $2$-cell is a bimodule map. The composition of $1$-cells is ...
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1answer
45 views

Why $M \otimes M$ does not have a ring structure?

I am reading some section about tensor algebras, and I don't have clear the idea on why $M \otimes M$ dont have a ring structure, where $M$ is an $R$-module. R is commutative and $1 \in R$. So far my ...
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0answers
29 views

Reference request: extending tensor product of modules

I'm looking for a reference to a construction similar to the following. I have a right R-module, $A_{K'}$, and a left R-module, $_KB$, where $K$ and $K'$ are fields and $K'\subset K$. I want to take ...
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1answer
106 views

How to calculate sum of combinations with different n and k

Input: $[X,Y]$ and $L$ Output : no of increasing sequence of length L and all elements should be $X\le i \le Y$ e.g: for $[6,7]$ and $2$ sequences are $6,66,67,7,77.$ For the above question my ...
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1answer
29 views

Two short exact sequences with projective objects in the middle

Problem: Prove that for two short exact sequences $$ 0\rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0 $$ $$ 0\rightarrow A' \xrightarrow{f'} B' \xrightarrow{g'} C \rightarrow 0, $$ ...
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1answer
47 views

How to prove tensor product is exact when acted on split short exact sequence?

I know tensor product is right exact, but I can't figure out why it's exact when it is acted on a split short exact sequence. In addition, can you give an example that tensor product acts on a short ...
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1answer
38 views

Period of a module vs element

I believe that my notes are a little off because it seems that in one instance per$(M)$ is a set and in another case it's an element. Maybe a computation from the notes would help clarify the ...
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1answer
48 views

Kernel of a map on tensor product of modules

Let $M,N, P$ be $A$-modules, and let $f:M \otimes N \to P$ be an $A$-homomorphism. If $m \otimes n \in \ker f$ implies $m\otimes n =0$ for all $m\in M, n\in N$, does it follow that $\ker f=0?$ For ...
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0answers
77 views

Property of free submodules for a module over a PID

It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is not free. We can take $R=M=K[x,y]$ , $L=<x>$ , ...
3
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1answer
42 views

How to define an exact sequence of $\mathbb Z$-modules?

Let $p$ be a prime. How to make an exact sequence of $\mathbb Z$-modules (abelian groups): $$0\longrightarrow \mathbb Z_p\stackrel{f}{\longrightarrow} \mathbb Z_{p^2}\stackrel{g}{\longrightarrow} ...
3
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2answers
39 views

On the existence of finitely generated modules with finite injective dimension

Assume $R$ is a commutative local Noetherian ring. It is known that if there is a finitely generated module with finite injective dimension then $R$ is Cohen-Macaulay. My question is: if $R$ is ...
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2answers
36 views

Is the sequence of $\mathbb Z$-modules below exact?

I came across the following example in my algebra textbook: Considere the sequence of $\mathbb Z$-modules: $$0\longrightarrow \mathbb Z_2\stackrel{f}{\longrightarrow}\mathbb ...
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1answer
20 views

Prove that if M is irreducible but not cyclic, then for every $r \in R$ and $m \in M$ we have $ rm = 0.$

Prove that if M is irreducible but not cyclic, then for every $r \in R$ and $m \in M$, $rm = 0.$ This is what i write but i don't know how to continue Suppose there exists an $r \in R$ and $m ...
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0answers
151 views

The dimension of vector space $F^{X}$.

Here's the problem: Let $F$ be field, $X$ an infinite set and $F^{X}$ be the set of all functions $f:X\rightarrow F$. Then $F^{X}$ is a vector space over $F$ (with $(f+g)(x)=f(x)+g(x)$ and ...
3
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1answer
45 views

Group ring C[Z/n] and Artin-Wedderburn decomposition

I am trying to answer the following questions, which I assume follow on from eachother each other; Write $\mathbb C$[$\mathbb Z$/n] as a product of simple rings. For abelian groups $G_1$, $G_2$, ...
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1answer
27 views

By Zorn's Lemma, a module can be finitely generated by its some elements and a submodule?

Take A as an R-module, and B is its submodule. We can get a submodule $B_1$ of A generated by B and an element $b_1 \in A\backslash$B. By the similar way we can get a submodule $B_2$ of A generated by ...
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1answer
18 views

Submodule of a finitely generated module over a Dedekind domain

If $R$ is a Dedekind domain then is a submodule of a finitely generated $R$-module also finitely generated over $R$? Many thanks in advance.
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1answer
21 views

A question about the quotient of two chain complexes of $R$-modules

Let $P$ be an acyclic object of $Ch_R$, let $P^{(k)}$ be the chain subcomplex of $P$ which agrees with $P$ above the degree $k-1$, contains $Bd_{k-1}P$ in degree $k-1$, and vanishes below degree ...
1
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1answer
35 views

$M\simeq D$ and $N\simeq D$ then $M\oplus N\simeq D\oplus D$?

Suppose $D$ is a PID and $M,N$ are $D$-modules and we have $\varphi_1:D\longrightarrow M$ and $\varphi_2:D\longrightarrow N$ isomorphism. Then if I take $\varphi:D\oplus D\longrightarrow M\oplus N$, ...
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1answer
36 views

Over a Bezout domain $R$, show every finitely generated submodule of $R^n$ is free

Let $R$ be an integral domain such that every finitely generated ideal of $R$ is principal. Show that every finitely generated submodule of $R^n$ is free. We know if $R$ is a PID, then the ...
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1answer
19 views

Different definitions of submodule

I have stumbled upon two different definitions of a submodule and I cant see why they are equivalent, let me state them ($M$ is the module and $A$ is the commutative ring with unity): First ...
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1answer
47 views

Does a module always have a cyclic non-zero quotient?

Let $M$ be a module over a ring $R$. My question is: Does there exists a submodule $N$ such that $M/N$ is isomorphic to $Ra$ for some $a\in M\setminus N$, that is, $M/N$ is 1-'dimensional'?
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1answer
24 views

$\text{Coker}(f),\text{Coker}(g)$ projective modules $\Rightarrow$ $\text{Coker}(gf)$ projective

Let $\textbf{Ch}_R$ the category of chain complexes of $R-$modules ($R$ is an associative ring with unit). I want to prove that this cat. satisfies the model category axioms. In particular we want to ...
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0answers
45 views

On semisimple rings

Let $R$ denote a ring with unity. I know that, if $R$ is semisimple, then every $R$-module is semisimple. In particular the class of indecomposable $R$-modules coincides with the class of simple ...
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1answer
61 views

Atiyah-MacDonald, 3.18. Why is $B_q$ a local ring of $B_p$?

This question is on the hint that the book gives to finish the exercise. Namely, if $f: A \rightarrow B$ a flat homomorphism of rings, $q$ a prime ideal of $B$ and $p = q^c$, then $B_q$ is a local ...
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0answers
11 views

Can the injective module map be seen as the composite of the injective maps like this?

Here A,B are R-modules, and the map f:A to B is injective. I wonder if I can see the map f as the composite of injective module maps $f_1f_2...f_n$, such that $f_1$ is the injective map from A to ...
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0answers
38 views

Specific examples of submodules.

I have two issues for which I seek solutions for a long time : 1- Submodules of finitely generated modules need not be finitely generated. I searched and i found that the submodule $K$ of the ...
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1answer
30 views

about homomorphismes of modules

My issue is the following: Let $M$ be an $A$-module with $A$ a commutative unitary ring, $u: M\longrightarrow M $ an $A$-morphism. I cannot solve or find the proof of the following assertions: 1- ...
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1answer
45 views

Properties of module length

Let $e_{A}(\phi, M): = l_A(\mathrm{coker}(\phi) ) - l_A(\ker(\phi))$. In my book it is stated that if $IM = 0 \implies e_{A}(\phi, M) = e_{A/I}(\phi, M)$ and this seems to be obvious for the author. ...
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3answers
166 views

Proving that two quotient modules are isomorphic

Given a ring $R$ and $R$-modules $A,B,C,D$ such that $$\sigma:A \rightarrow B, \tau: C \rightarrow D, \rho: A \rightarrow C, \kappa: B \rightarrow D, \ \mathrm{and} \ \kappa \circ \sigma = \tau \circ ...
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1answer
50 views

Prove that $Ker(g \otimes k)= Im(f \otimes 1_{N}) + Im (1_{M} \otimes h)$

Suppose we have two short exact sequences: $$0 \to M' \mathrel{\overset{f}{\to}} M \mathrel{\overset{g}{\to}} M'' \to 0 $$ in Mod-R $$0 \to N' \mathrel{\overset{h}{\to}} N \mathrel{\overset{k}{\to}} ...
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1answer
39 views

Proving that a module can be decomposed as a direct sum of submodules

Letting $X$ be a ring and $K$ be an $X$-module, I need to show that if $K \cong A \times B$ for some $X$-modules $A,B$, then $\exists$ submodules $M'$ and $N'$ of $K$ such that: $K=M' \oplus N'$ $M' ...
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0answers
31 views

Does a finite generated submodule with another element always generate a finite submodule?

Let M be a module and N be a finite generated submodule of M, and x$\in$M\N, then consider the existence of module B which is generated by N and x. I think, as N is f.g thus we can see N as ...