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

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1answer
29 views

Does the usual filtration on graded objects satisfy a universal property?

Let $\mathsf{A}$ be an abelian category, such as $R \mathsf{Mod}$ for concreteness. We can think of the category of $\mathbb{Z}$-graded objects in $\mathsf{A}$ as the functor category $\mathsf{A}^\...
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3answers
47 views

Writing $\left(\mathbf Z/2\oplus\mathbf Z/4\oplus\mathbf Z/8\right)\oplus\left(\mathbf Z/9\oplus\mathbf Z/27\right)$ in a different way

Writing $\left(\mathbf Z\big/2\oplus\mathbf Z\big/4\oplus\mathbf Z\big/8\right)\oplus\left(\mathbf Z\big/9\oplus\mathbf Z\big/27\right)$ in a different way If $M$ is equal to above how can I find ...
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1answer
13 views

Cokernel of a module homomorphism

Let $A$ a $K$-algebra. Let $M$, $N$ $A$-modules and $f:M\rightarrow N$ a module homomorphism. The cokernel of $f$ is $Cokerf=N/Imf$ I define a homomorphism $\rho:N\rightarrow N/Imf$ by $\rho(n)=n+Imf$....
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1answer
33 views

Difference between $\mathrm{ass}(M)$ and $\mathrm{supp}(M)$

I'm reading through Lang's chapter about Noetherian modules and rings in Algebra. More specifically, subchapter about associated primes. Let $A$ be a commutative ring, and $M$ its module. If $\...
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0answers
38 views

Question on projective modules, a course in homological algebra

Show that if $0\rightarrow N\xrightarrow{\tau} P\xrightarrow{\epsilon} A\rightarrow 0$ and $0\rightarrow M\xrightarrow{\tau'} Q\xrightarrow{\epsilon'} A\rightarrow 0$ are exact sequences with $P,Q$ ...
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1answer
26 views

Counterexample: Homology with Local Coefficients versus Homology with Module Coefficients

Definitions Let $X$ be a nice space with universal covering $\widetilde{X}$. Homology $H_*(X,R)$ with Ring Coefficients $R$ is the homology of \begin{align*} C_n(X;R) :=&\; \text{ free left $R$-...
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1answer
15 views

Calculating the modulus

I have a function $z=|\frac{-4i\pm2i\sqrt3}{2}|$ Now I would simplify this by doing: $|\frac{-4i+2i\sqrt3}{2}|=|-2i+i\sqrt3|=|-2+\sqrt3|=2+\sqrt3$ However the book I am using gives: $|\frac{-4i+2i\...
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1answer
21 views

Construction of a Module isomorphism

Let $R$ be a PID. Consider the sets $X_0=\{v_0,v_1,v_2\}$ and $X_1=\{e_1,e_2,e_3\}$ and let $C_i$ be the free $R$-module on $X_i$ for $i=0,1$. Consider the $R$-module homomorphism $$C_1\;\...
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2answers
208 views

When does tensor product have a (exact) left adjoint?

Let $A$ be a commutative Noetherian ring, and let $F$ be a flat $A$-module. We can assume $A$ is local, so $F$ is projective. Question 1. When does $F\otimes_A-$ preserve injective objects? ...
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1answer
36 views

What does the $(m,n)$ mean in context of $\mathbb{Z}_{(m,n)}$

On the section about the Hom sets of modules, Hungerford has an exercise that asks to show that $$\operatorname{Hom}(\mathbb{Z}_m, \mathbb{Z}_n) \cong \mathbb{Z}_{(m,n)}$$ and then in the next ...
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1answer
23 views

Constructing $R$-modules using formal sums

Let $X=\{x_1,\cdots,x_n\}$ be a finite set and let $R$ be a ring. We construct an $R$-module $M$ in the following way: an element $a$ in $M$ is a "formal sum" $a=\sum_ir_ix_i$ where $r_i $ are in $R$. ...
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0answers
40 views

equivalence of categories between modules and vector space

Let $Map_k$ the category where the objects are triples $(V,W,f)$, where $V$ and $W$ are finite dimensional $K$-vector spaces and $f:V\rightarrow W$ is a $K$-linear map. A morphism from $(V,W,f)$ to $(...
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1answer
15 views

Classify all simple modules over $k$ and $k[x]$.

A module is said to be simple if it is nonzero and has no nontrivial proper submodules. Let $k$ be a field. Classify up to isomorphism all simple modules over $k$ and $k[x].$ Since a simple module ...
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4answers
44 views

On the kernel of a certain module epimorphism $\mathbb{Z}^2 \to \mathbb{Z}/6\mathbb{Z}$

In order the construct a certain projective resolution of $\mathbb Z / 6 \mathbb Z$ I need to find the kernel of the ($\mathbb Z$-) module morphism: $$\epsilon_0 : \mathbb Z^2 \to \mathbb Z / 6 \...
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1answer
44 views
+100

Modules that allow an infinite exact sequence on them

I'm looking for a characterization of modules that fit the following property: (*) There's an infinite exact sequence with $\phi_i \neq 0 \space \forall i\in I $ $\require{AMScd}$ \begin{CD}\cdots @&...
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0answers
23 views

On the cokernel of a module mapping

Given two modules $A,B$ on the same ring $\Lambda$, consider an homomorphism $f:A\longrightarrow B$. Then we can define: $$\operatorname{Ker}(f)=\left\{a\in A\ |\ f(a)=0\right\}$$ $$\operatorname{Im}(...
2
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1answer
44 views

Matrices representing injective homomorphisms

Let $R$ be a ring and $M$, $N$ finitely generated free modules modules over $R$. Let $A$ be a matrix representing a homomorphism $f: M \rightarrow N$. We know that the map $f$ is injective if and only ...
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3answers
396 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 ...
2
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0answers
24 views

free modules being isomorphic with different dimension

This question is from the book A course in Homological algebra by Hilton and Stammbach Let $V$ be a vector space of countable dimension over the field $K$. Let $\Lambda=\rm{Hom}_K(V,V)$. show that, ...
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1answer
43 views

Isomorphic for free modules implies for projective [closed]

I have if $F$ is a free $R$-module then $\mathrm{Hom}(L,M) \otimes F$ is isomorphic to $\mathrm{Hom}(L,M \otimes F)$. Then how can I conclude that $\mathrm{Hom}(L,M) \otimes P$ is isomorphic to $\...
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1answer
58 views

Is there a theory of generalized eigenvectors over commutative rings?

Brown's Matrices over Commutative Rings book discusses the theory of eigenvalues, eigenvectors, and diagonalizing matrices over commutative rings, but unless I've missed something, nothing like ...
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3answers
46 views

Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
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1answer
25 views

Property of free modules

Question is regarding following property of free modules: Let $P$ be a free $R$ module. To every surjective homomorphism $f:B\rightarrow C$ of $R$ modules and to every homomorphism $g:P\...
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2answers
26 views

Help with proving a fact about injective modules

Let $R$ be a ring with identity. I am trying to show that an $R$-module $A$ is injective if and and only if for every left ideal $L$ of $R$ and every $R$-module homomorphism $g: L \rightarrow A$, ...
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1answer
27 views

A question on extension of rings which related to their direct summands

I read "Foundations of Module and Ring Theory" of Robert Wisbauer and I got stuck in this problem: *Show for a ring $R$. The following assertions are equivalent: (a) $R$ has a unit. (b) If $R$ is ...
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0answers
35 views

Question about generators and Hom functor

In a given category $\mathcal{C}$ I want to prove the following statement: If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{...
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0answers
12 views

Indecomposable Modules over an algebra

If I have $K$ subfield of $F$ and $A$ an $K-$algebra (associative ,finite dimensional and with with unity). And take an ciclic indecomposable $A-$module $M=A\lambda$. Then $A\otimes_K F(\lambda\...
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0answers
29 views

finitely generated projective R-modules

Let $L$, $M$ and $N$ be $R$-modules. Then I know that there is a natural homomorphism from $Hom(L,M) \otimes N \to Hom(L,M \otimes N)$ defined by $f \otimes n \to \tilde{f}$ where $\tilde{f}(\ell)=f(\...
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0answers
21 views

What is number of irreducible characters in modular representation?

In ordinary representation, I know that the number of irreducible characters is the number of conjugacy classes, what about the modular representation? Can I find a good and simple book on modular ...
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0answers
14 views

Application of Structure Theorem to Prove Simultaneous Diagonalizability and Group of Units of Cyclic Groups

I am reading these notes on Modules over PID. Exercise 67 (pg 24) asks to prove that: Problem. Let $A$ and $B$ be $n\times n$ matrices with complex entries. Then $A$ and $B$ are simultaneously ...
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1answer
36 views

Proving that a prime ideal $p \subset R$ yields a prime ideal $p[x] \subset R[x]$

I'm curious as to whether I can have my proof critiqued. Proposition : Let $\mathfrak{p}$ be a prime ideal in a ring $R$. Show that $\mathfrak{p}[x]$ is a prime ideal in $R[x]$. Proof : Suppose $\...
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1answer
31 views

The sum of a family of submodules

I'm just reading Atiyah-Macdonald's commutative algebra text, specifically the section on Operations on Submodules. I just have some questions regarding constructions. Firstly, suppose $M$ is an $A$-...
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1answer
22 views

Polynomial ring as direct sum of modules

I'm aware that this topic has been posted previously. However, I'm not wanting to prove anything. I just want to understand, somewhat intuitively, why $R[x]$ as a module over $R$ is given by $$R[x] =\...
2
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1answer
29 views

Compatibility of Yetter-Drinfeld modules.

Let $H$ be a Hopf algebra. A Yetter-Drinfeld module over $H$ is a triple $(V, \cdot, \delta)$, where $\cdot : H \otimes V \to V$ , $\delta : V \to H \otimes V$ are actions and coactions respectively, $...
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1answer
49 views

Is it $\prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i}$ true? [closed]

Suppose that $A$ is a ring and $S$ is a multiplicative closed subset of $A$. I wonder that do we have $\prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i}$? Can you give me ...
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1answer
194 views

Why does this ring have rank $k!$?

Let $R$ be any ring free of rank $k$ over $\mathbb{Z}$ having non-zero discriminant. Let $R^{\otimes k} = R \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} R$. Then $R^{\otimes k}$ is a ring of rank ...
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3answers
461 views

Prove that $R \otimes_R M \cong M$

Let $R$ be a commutative unital ring and $M$ an $R$-module. I'm trying to prove $R \otimes_R M \cong M$ but I'm stuck. If $(R \otimes M, b)$ is the tensor product then I thought I could construct an ...
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2answers
232 views

Showing that if $R$ is a commutative ring and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$.

Let $R$ be a local ring, and let $\mathfrak m$ be the maximal ideal of $R$. Let $M$ be an $R$-module. I understand that $M \otimes_R (R / \mathfrak m)$ is isomorphic to $M / \mathfrak m M$, but I ...
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2answers
792 views

How to prove that $R/I \otimes_R M \cong M / IM$ [duplicate]

Possible Duplicate: Showing that if $R$ is local and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$. In one of the answers to one of my previous questions the ...
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1answer
32 views

What is the simplest example of the tame representation type?

What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ...
2
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1answer
35 views

Hom and direct sums 3

Let $\{M_i\}_{i \in I}$ and N be left R modules where R is not necessarily commutative. Then how can we prove that $Hom_R(N, \bigoplus_{i \in I} M_i)$ is isomorphic to $\bigoplus_{i \in I}Hom_R(N,M_i)...
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2answers
26 views

About the definitions of direct product and direct sum of modules.

Direct product of modules can contain infinitely nonzero elements, but direct sum of modules must contain finitely nonzero elements. What's the point of the definitions? Why the definitions are ...
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0answers
21 views

If $N$ is finitely generated, then $(L :_{R} N)^{e}= (S^{-1}L :_{S^{-1}R} S^{-1}N)$ [duplicate]

I have question in this lemma. Please help me explain it more. Let $L$, $N$ be submodules of a module $M$ over a commutative ring $R$ and let $S$ be a multiplicatively closed subset of $R$. If $N$ is ...
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2answers
43 views

How many solutions of $\mod 63$ : $x^2=1 \pmod7 $ and $x^3=1\pmod 9$ [closed]

How many solutions $\mod 63$ , we have for: $$x^2=1 \pmod 7$$ and $$x^3=1 \pmod 9$$ Need to find them also.
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Rings and modules-ideals and submodules

I am taking a course in Commutative Algebra and the following lemma in a section on localisation raised some questions. Lemma: Let M be an R-module. The following are equivalent. (1) $M=0$ (2) $M_P=...
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1answer
224 views

Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
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0answers
22 views

$f:A\to B$ flat homomorphism of rings, $Q$ is a prime ideal of $B,$ and $P=Q^c.$ Why is $B_Q$ a local ring of $B_P?$

I have a question about Exercise 2.18 on Introduction to Commutative Algebra by Atiyah and MacDonald. Let $f:A\to B$ be a flat homomorphism of rings, let $Q$ be a prime ideal of $B$ and let $P=Q^c,$ ...
4
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1answer
105 views

Sum of free submodules of 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 $R=M=K[x,y]$ , $L=\left<x\right>$ , ...
2
votes
1answer
26 views

Is the tensor algebra functor a strong monoidal functor?

Let $K$ be a commutative ring. Is it true that $T(V \oplus W) \cong T(V) \otimes_K T(W)$ as $K$-algebras for any $K$-modules $V$ and $W$? The reason I ask is that I have heard it mentioned (I don't ...
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0answers
21 views

When does the unit of universal enveloping algebra adjunction fail to be injective?

The Poincaré–Birkhoff-Witt (PBW) theorem implies that if $K$ is a commutative ring and the Lie $K$-algebra $\mathfrak{g}$ is a free $K$-module, then the unit $\eta_{\mathfrak{g}}: \mathfrak{g} \to U (\...