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

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74 views

Determine a basis for $\mathbb{Z} \oplus \mathbb{Z}$ which determines a basis for the submodule $N$ generated by $(6,9)$

Proof Clearly the rank of $\mathbb{Z} \oplus \mathbb{Z}$ is $2$, so we must have that the rank of $(6,9)$ is $\leq 2$.Let $e_1 = (1,0)$ and $e_2 = (0,1)$ be a basis for $\mathbb{Z} \oplus ...
3
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1answer
53 views

Length of tensor product of finite length modules is finite

Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M\otimes_R N$ has finite length, and $l(M\otimes_R N) \le l(M)l(N)$. I know the question has been posted ...
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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
60 views

$R/Ra$ is an injective module over itself

Let $R$ be a PID, $a\in R$ be a nonzero nonunit in $R$. Prove that $R/Ra$ is an injective module over itself. If $R$ is a PID, every $R$- divisible module is injective, but the question concerns ...
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1answer
16 views

Isomorphism of tensor product involving a principal ideal

This question arose when dealing with a long exact sequence of Tor. Let $R$ be a (not necessarily commutative) ring, $g$ a central element of $R$ and $M$ a right $R$-module. We have an exact sequence ...
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2answers
43 views

Different module on same set

Let $M$ be a (left) $R$-module. Is there any example that $M$ has two $R$-module structure? (i.e. has different $R$-multiplications)
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2answers
41 views

Factorization in modular arithmetic

Is this expansion a legal step? $12^8\mod15 = 2^8 * 2 ^ 8 * 3 ^ 8\mod15$
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0answers
39 views

Property of free submodules for a module over a PID [duplicate]

This question was asked here and remains without solution. 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 ...
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0answers
15 views

Extra Operation required for Smith Normal Form over PID-Theoretical Justification

Why does one need an extra operation for performing smith normal form over a PID? One might suspect and say that it is because of the lack of Euclidean algorithm or just say that we need the ...
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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
40 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 ...
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1answer
73 views

Dual of polynomial ring

Consider the free $k$-algebra $k[x_i]_{i \in I}$ indexed by $I$. Then is $Hom_{k-Mod}(k[x_i]_{i \in I},k) \cong k[x_i]_{i \in I}$?
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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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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 ...
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1answer
23 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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129 views

Why has the category of all discrete $G$-modules not enough projectives when $G$ is profinite?

I found this article about Galois Cohomology. In it, it says that, when $G$ is a profinite group and $\mathbf{C}_G$ is the category of all discrete $G$-modules, $\mathbf{C}_G$ doesn't have enough ...
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1answer
65 views

$M\times N$ Doesn’t Have a Module Structure

In Keith Conrad's notes (page 4) is written: For two $R-$modules $M$ and $N$ , $M\oplus N$ and $M\times N$ are the same sets, but $M\oplus N$ is an $R-$module and $M\times N$ doesn’t have a ...
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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
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
59 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
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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2answers
99 views

Direct sum of indecomposable modules

Prove or disprove: There exist indecomposable modules $V,U,W$ such that $\displaystyle{ V \oplus U \cong V \oplus W }$ and $\displaystyle { U \not \cong W}$. I think that this is true, but I have no ...
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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 ...
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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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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
58 views

Identifying the tensor product of two given modules over $\mathbb Z/2\mathbb Z$.

Suppose that $\pi$ denotes the finite ring of order $2$, $\mathbb Z_+$ the trivial $\pi$-module and $\mathbb Z_-$ the non-trivial $\pi$-module (i.e., the generator acts by multiplication by −1). I ...
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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
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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2answers
110 views

Decomposition of finitely generated graded modules over PID

I found this decomposition theorem used in a paper I'm reading, but it isn't referenced and I can't seem to find it in any of the books I have: Every graded module $M$ over a graded PID decomposes ...
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1answer
43 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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1answer
91 views

Are all simple left modules over a simple left artinian ring isomorphic?

I've a basic question. $R$ is a simple left artinian ring. I want to show that all simple left $R$-modules are isomorphic. A simple $R$-module $M$ is isomorphic to $R/J$ where $J$ is a maximal ...
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2answers
163 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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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
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
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
70 views

Classifying torsion-free injective modules over a PID

So the question is as in the title: Let $R$ be a PID. Classify all torsion-free injective modules. I know that it is going to be divisible, and using torsion free, if we define ...
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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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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 $R=M=K[x,y]$ , $L=<x>$ , ...
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34 views

What is interesting (useful) about Multiplicity?

Multiplicity is defined at 4.1.5; Bruns_Herzog. People say it is an important invariant. I don't know what idea is behind this definition and What is interesting/useful about it. what important ...
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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
43 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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150 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 ...
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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 ...