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

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46 views

Is $h^*$ injective in this case?

Let $N$ and $N'$ be finite rank free $\mathbb Z$-modules. Let $M=\operatorname{Hom}_{\mathbb Z \text{-mod}}(N,\mathbb Z)$ and $M'=\operatorname{Hom}_{\mathbb Z \text{-mod}}(N',\mathbb Z)$ . Suppose ...
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2answers
62 views

Can you prove that over a division ring , every module is both projective and injective?

I have problem with this : over a division ring or a ring with identity every module is both projective and injective . Can you help me? thank you .
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2answers
39 views

Basis of the free $R$-module $R[x]/(f)$ [closed]

If $f\in R[x]$ is any monic polynomial of degree $n$, then $R[x]/(f)$ has basis $\{1,x,…,x^{n−1}\}$. Can someone explain this?
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1answer
26 views

On the properties of submodules

Let $R$ be a ring with unity and $N$ be a left $R$-module. How to prove that if $K+M=L+M$ and $K\cap M=L\cap M$, then $K=L$? Where $K,L,M$ are left $R$-submodules of $N$ s.t $K\subseteq L$.
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0answers
49 views

associative and commutative F-algebra with basis

let $G$ be a finite-dimensional $F$-algebra and let $X=\{x_1,...,x_m\}$ be the basis. The structure constants of $G$ with respect to the basis are the scalars $f_{rsk}\in F$ for $1\leq r,s,k\leq m$ ...
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0answers
25 views

Decompositions via Invariant Subspaces

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$, and let $T: V \rightarrow V$ be a linear transformation such that $T^{4} - I = T^{3} + 3T^{2} + T + 3 =0$. Prove that $V$ is a direct ...
4
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0answers
41 views

Finitely generated torsion module over a Dedekind domain

Let $M$ be a finitely generated torsion module over a Dedekind domain $R$. Show that there exist nonzero ideals $I_1 \supseteq \cdots \supseteq I_n$ of $R$ such that $M \cong ...
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1answer
19 views

endomorphism of the right R-module $eR$

Let $e$ be a non-zero idempotent of a ring R, I want to find the unique endomorphism of the right R-module $eR$ such that $f(e)e=ere$ for a fix $r\in R$. I tried to solve this problem and I have found ...
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0answers
11 views

$\text{Hom}$ to a projective $D[G]$-module for a complete DVR $D$

Suppose you have a complete DVR $D$ and a finite group $G$ with $D[G]$-modules $A$ and $B$. Does $B$ being projective imply that $\text{Hom}_{D[G]}(A,B)$ is $D$-free? Or should it be $A$ that's ...
2
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1answer
17 views

Proof there is a basis of $V$ w.r.t. which $T$ is the companion matrix of $a(x)$?

Given a linear transformation $T$ on a finite-dimensional vector space $V$ over a field $K$, and giving $V$ the $K[x]$-module structure determined by $f(x)v = F(T)(v)$, Frederick Goodman shows on page ...
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1answer
18 views

$\zeta:eM\to eR\otimes_R M$ is a group isomorphism

Let $R$ be a ring, $e\in R, e\neq0$ be an idempotent element (i.e. $e^2=e$). Let $_RM$ be a left $R$-module. I have to show that $$ \zeta:eM\to eR\otimes_R M $$ definded by $em\mapsto e\otimes m$ is ...
1
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1answer
50 views

Why is $V$ isomorphic to $F[x]/(b(x))$?

Let $V$ be a finite-dimensional vector space over $F$, and let $T\colon V\to V$ be a linear transformation. Consider $V$ as an $F[x]$-module where the action of $x$ on $V$ is defined as the action of ...
3
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2answers
27 views

Is this matrix ring semisimple?

I'm trying to solve the following problem. Let $\mathbb{C}$ be the complex numbers and $\mathbb{C}[x]$ the polynomial ring over $\mathbb{C}$. Let $\alpha\in\mathbb{C}$. For which $n\geq 1$ is the ...
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0answers
14 views

Division algebras and idempotents

Suppse that A is a finite dimensional algebra and R be its radical. Prove the follwing statement: If the quotient algebra A/R has no non-trivial idempotent, then A/R is division algebra? I would be ...
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0answers
32 views

Calculating an endomorphism ring of a module over $R = \Bbbk[x,y,z]/(xy-z^2)$

Let $R = \Bbbk[x,y,z]/(xy-z^2)$, and let $M = (x,z)$. I want to determine $\text{End}_R(M)$. Since $R$ is a domain, I think that we're able to identify $\text{End}_R(M)$ with $ \{ q \in Q(R) \mid qM ...
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0answers
19 views

Caracterization of pure submodule by Hom functor

Let $M$ be an $R-$module and $N$ is a submdule of $M$. $N$ is said to be a pure submodule of $M$ if the sequence $$0\rightarrow A\otimes N \rightarrow A\otimes M$$ is exact for every $R-$module $A$. I ...
1
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1answer
21 views

Definition for non-degenerate module

[QUESTION] If $R$ be a ring, what is the meaning of a non-degenerate $R$-module? In a previous question post at (What is a non-degenerate module?), some experts said that if $M$ is a $R$-module such ...
2
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1answer
29 views

Realizing a quotient ring as an abelian group

Show that $\mathcal{M} = \mathbb{Z}[X] /(X^{2} - X, 4X +2)$ is finitely generated abelian group. What is the general procedure for handling these types?
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0answers
76 views

$\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is a stable C*-algebra

Let $B$ be a C*-algebra. I want to prove that $\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is stable, that is, $\mathcal{K} \otimes B$ is isomorphic to $B$ as ...
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1answer
41 views

Distinguished category of groups, is it abelian?

Let $\mathcal{A}$ denote the following category. The objects of $\mathcal{A}$ are the pairs $(H,G)$ where $G$ is an abelian group and $H$ is a subgroup of $G$. The morphisms between $(H,G)$ and ...
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0answers
30 views

Dimension of subspace of commuting matrices

I was hoping for some confirmation of a proof to the following preliminary exam question: Fix an $n \times n$ matrix $A$ with entries in an algebraically closed field $k$. Let $C$ be the space ...
0
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1answer
39 views

$\tau: S^{-1} (\operatorname{Hom}_A(M, N)) → \operatorname{Hom}_{S^{-1}A}(S^{-1}M, S^{-1}N)$ is injective if M is a finitely generated A-module

Let A be a ring. Let M and N be two A-modules, S is a multiplicatively closed subset of A. Show that there is a homomorphism of $S^{-1}A$-modules $$\tau: S^{-1} (\operatorname{Hom}_A(M, N)) → ...
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1answer
85 views

B is a flat A-module, and M is a finitely generated $A$-module then $σ:\operatorname{Hom}_A(M, N)⊗_AB→\operatorname{Hom}_B(M⊗_AB,N⊗_AB)$ is injective

Let A and B be rings, and let f : A → B be a homomorphism of rings; we consider B as an A-module with the structure induced by f. Let M and N be two A-modules. There is a map of $B$-modules ...
1
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2answers
42 views

What does $\mathfrak{m}(\varprojlim R/\mathfrak{m}^i)$ mean?

Let $\mathfrak{m}$ be an ideal of the ring $R$, and consider it as an $R$-module. What does $\mathfrak{m}(\varprojlim R/\mathfrak{m}^i)$ mean? In the limit the morphisms are the natural projections. ...
4
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1answer
39 views

Coordinate-free definition of elementary divisors

There is a general mantra in math which says that what is independent of bases shall be defined independent of bases. Well, it is well known that the elementary divisors of a linear map ...
0
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1answer
33 views

$\text{Hom}_k(M,N)\cong M^*\otimes_k N$ as Hopf-algebra modules.

I'm reading Representations and Cohomology by D.J. Benson. At the beginning of the third chapter the following is explained: Let $\Lambda$ be a bialgebra over $R$ and $M,N$ left $\Lambda$-modules. We ...
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1answer
24 views

One dimensional simple modules

Let $A$ be a finite dimensional algebra over an algebraically closed field and let $mod A$ denote the category of finite dimensional A-modules. Problem Suppose that every simple $A$-module is ...
2
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1answer
26 views

When does non-spanning imply linear independence in a free $R$-module

In a free $R$-module $M$, a set $\{a_1,...,a_k\} \subset M$ is linearly independent if and only if whenever $\sum r_ia_i=0$ with $c_i \in R$, it follows that for every $i$, $c_i=0$. Now suppose the ...
0
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0answers
19 views

Show that there exists an $R-submodule$ $M_2$ such that $M$ isomorphic to $M_1 \oplus M_2$

Let $R$ be the group ring of a finite group $G$ over a field $K$. Suppose that the characteristic of $K$ does not divide $|G|$. Let $M$ be an $R-module$ and $M_1$ be a $R-submodule$ of $M$. Show that ...
1
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1answer
26 views

Skeleton of finitely generated modules

If $A$ is a pid, then let $C$ be the category of finitely generated $A$-modules. Is there a skeleton of $C$ that can be described explicitly? I imagine that the structure theorem can be used to do ...
1
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1answer
42 views

Why should one show that $x$ and $0$ are linearly dependent?

In exercise 1 of D&F section 12.1, it's stated: Let $M$ be a module over the integral domain $R$. Suppose $x$ is a nonzero torsion element in $M$. Show that $x$ and $0$ are "linearly ...
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1answer
15 views

Understanding Jurgen Neukirch proof that $A[b_1, \dots, b_n]$ finitely generated $\implies$ it's integral over $A$.

... Conversely, assume that the $A$-module $A[b_1, \dots, b_n]$ is finitely generated and that $w_1, \dots, w_r$ is a system of generators. THen for any element $b \in A[b_1, \dots, b_n]$, one ...
0
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1answer
14 views

$b w_i = \sum\limits_{i=1}^r a_{i,j} w_j \implies \det(bI - (a_{i,j})) w_i = 0$ in an $R$-module.

Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$ module. Then if $\{w_j : j = 1 \dots r\}$ is a set of generators for $M$ and $b \in R$, then $b w_i = \sum\limits_{j=1}^r ...
-1
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1answer
46 views

M is finitely generated but $\ker(f)\subset M$ is not. [duplicate]

So I'm searching for an homomorphism of $A$-modules $f:M\mapsto N $ and some A-modules $M$, $N$ such both are finitely generated but $\ker f\subset M$ is not. Thanks.
0
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1answer
12 views

For epimorphism $\theta:P \rightarrow M$, where $Ker\theta \ll P$, then if $P'\leqslant P$, $\theta(P')=M \Rightarrow P'=P$?

I am reading into projective covers of modules and this statement arose after the definition, but I am not entirely sure how to show the statement to be true. (In context, $P$ is the projective cover ...
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0answers
25 views

Can a module have an infinite number of compositions series?

Is there an example of a module that has an infinite number of composition series? I would think not, if there is it would have to have an infinite number of submodules.
4
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1answer
43 views

Left adjoint of a strange functor

I'm looking at the category (call it $\mathcal{C}$) of colimit preserving functors from $R$-mod to $Vect$, where $R$ is some commutative $k$-algebra. There is an obvious functor $$ \mathcal{F} : ...
2
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1answer
28 views

Coprime submodules: Quotients, intersections and direct sums

Let $R$ be a unital ring, and let $M$ be an $R$-module. Suppose $M_1, \ldots, M_n$ are $R$-submodules of $M$ such that $M_i + M_j=M$, for $i \neq j$. Are the modules $M/(\cap_{i=1}^n M_i)$ and ...
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1answer
54 views

$f:A\to B$ epimorphism if and only if $B/f(A)$ is a torsion module [closed]

I want to prove the double implication: $f: A\to B$ is an epimorphism in the category of torsion-free abelian groups $\Longleftrightarrow$ $B/f(A)$ is a torsion $\mathbb Z$-module. Thanks.
4
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2answers
144 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? ...
6
votes
1answer
82 views

Tensor product of modules: $\Bbb Z[x]/\langle f(x) \rangle \otimes_{\Bbb Z} \Bbb Z/p\Bbb Z \cong (\Bbb Z/p\Bbb Z)[x]/\langle f(x) \rangle$

This is a question about tensor product of modules. How to show that $$\Bbb Z[x]/\langle f(x) \rangle \otimes_{\Bbb Z} \Bbb Z/p\Bbb Z \cong (\Bbb Z/p\Bbb Z)[x]/\langle f(x) \rangle$$ for any ...
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1answer
34 views

Complexification of $\mathbb{Z}$ using tensor products

I would just like a brief explanation of how to evaluate, the following tensor product when the vector space $V$ is replaced by the set of integers $\mathbb{Z}$. $$ ? = V\otimes_{\mathbb{R}} ...
0
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1answer
66 views

Isomorphism between $A$ and $B$ if $A/kA$ and $B/kB$ have same order? [closed]

Let $A$ and $B$ be finitely generated abelian groups, where for all $k\geq1$ the orders of $A/kA$ and $B/kB$ are equal. Are $A$ and $B$ necessarily isomorphic? Why (not)?
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1answer
79 views

Surjective endomorphism of abelian group is isomorphism

Let $A$ be a finitely generated abelian group and $f:A\rightarrow A$ a surjective homomorphism. How do I prove that $f$ is an isomorphism? And if $f$ were injective instead of surjective would the ...
2
votes
0answers
63 views

projective dimension over local ring

My question is as below. Let $(R,m)$ be a local ring such that $m$ consists of zero divisors. Prove that if $A$ is a finitely generated R-module, then $pd_R(A)$, the projective dimension of $A$ over ...
0
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1answer
33 views

Classify all direct summands of $K[X]\bigoplus K[X]$

Consider $M=K[X]$ as a module over itself. Now the claim I want to prove is that all direct summands of the module $N=M\bigoplus M$ are of the form $U_{f,g}=\{(hf,hg)\in N:h\in M\}$ for two fixed ...
1
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0answers
61 views

prove that tensor product is pushout

I want to show that if $A$ and $B$ are $C$-algebras (all rings are commutative) and $\varphi:A \rightarrow A \otimes_C B$ and $\phi:B \rightarrow A \otimes_C B$ are natural maps, then $A \otimes_C B$ ...
3
votes
1answer
86 views

Finding an explicit isomorphism from $\mathbb{Z}^{4}/H$ to $\mathbb{Z} \oplus \mathbb{Z}/18\mathbb{Z}$

There was a past qualifying exam problem, I was having trouble with, it is stated below as follows: In the group $G= \mathbb{Z} \times \mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}=\mathbb{Z}^{4}$, ...
1
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1answer
13 views

nonsimple module homomorphism

Let $A$ be an algebra with unity and $M$ be a left $A$-module. If $x$ is a nonzero element in $M$, then $Ax$ is a nonzero left $A$-module. If we define a map $f$ from the regular module $A$ onto $Av$ ...
8
votes
1answer
132 views

$A$-module is free if and only if equation involving Hilbert-Poincaré series holds.

Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector spaces $A_i$ and $M_j$ are ...