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

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0answers
15 views

Equivalence of definitions of minimal polynomial of a linear transformation $T$ over a vector space

Let $V$ be a vector space over $K$ and let $T\in End_{K}(V)$. Then $V$ has a $K[x]$-module structure via the operation defined as $f(x)v=f(T(v))$. Using this we can know a lot about the structure of ...
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0answers
22 views

Exterior power of a free module [duplicate]

Suppose $M$ is a free R module where $R$ is commutative and unital. Is $\Lambda^n M$, the nth exterior power of $M$ free?
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1answer
37 views

Additivity of direct limits

I am trying to understand the proof of a Lemma from Greenberg's Algebraic Topology. It says: Let $I$ be a directed set and $M_i$ a family of modules. Suppose that for every $i\in I$, we have ...
0
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1answer
39 views

Isomorphism between sum of submodules and their direct sum

Let $A$ be a commutative ring, let $M$ be an $A$-module, and let $M'$ and $M''$ be submodules of $M$ such that $M' \cap M'' = \{0\}$. Prove that $M' + M'' \cong M' \bigoplus M''$. I believe ...
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1answer
14 views

If $P/I_kP$ are finitely-generated, is it true that $P/IP$ is finitely-generated where $I=\bigcap I_k$?

I'm looking into some old results on "big projectives'', and trying to understand some steps. Assume that $R$ is a (commutative) ring and $I_1,\ldots,I_n$ are ideals. Let $I=I_1\cap\cdots\cap I_n$ be ...
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1answer
47 views

Image is zero in every module homorphism [on hold]

Suppose $R$ is a commutative ring, $A$ an $R$-module, and $A'$ is an $R$-submodule of $A$ such that for every $f \in \mathrm {Hom}_R(A,A')$ we have $f(A)=0$. Will this imply $A'=0$? I am not ...
3
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1answer
34 views

Can somebody explain me this proof on Kasch book (Modules and rings)?

I have a question about one step in the proof of Proposition 13.2.6: If $R_R$ is injective and ${_R}R$ is noetherian then $R_R$ is a cogenerator and $R$ is artinian on both sides. In the proof of ...
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1answer
36 views

Prove $\dim(V) \leq 2$

Consider $V$ a vector space over a field $K$. Consider a linear map $T:V\to V$ such that $T^2 = T$. Suppose $V$ is a cyclic $K[x]$-module. Prove that $\dim(V) \leq 2$. I know that the vector ...
4
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1answer
50 views

Ascending chain - Isomorphism theorems

So that what I did so far : Interesting results : $f$ is injective $\iff$ $\ker f = \{0\}$ $\ker f$ is a submodule of $X$ As $X$ is noetherian, then each ascending chain $M_1 \subset M_2 \subset ...
2
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1answer
36 views

Free Modules of Rank n

I'm prepping for prelim exams and realized I have a fundamental gap in understanding of free modules. I'm trying to answer the following question: Let $M = \mathbb{Z}^n$ and denote by $pM = ...
4
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1answer
55 views

Modules over ring of polynomials in several indeterminates.

I read this proposition and its proof in the Dummit's book (It's not exactly like this, but this is the idea of the proposition). Let $F$ be a field, $V$ a vector space over $F[x]$ and $T\in ...
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1answer
28 views

The order ideal of a finitely generated module over a Dedekind domain

Let $R$ be a Dedekind domain and $M$ be a nonzero finitely generated torsion $R$-module. In Curtis and Reiner's Methods of Representation Theory it states that $M$ has a composition series and if its ...
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2answers
37 views

Tensor algebra with semidirect product

I am having trouble parsing the following problem: Let $F$ be a field, $A$ an $F$-algebra with 1 (not necessarily commutative) and let $M$ be an $F$-vector space which is also an $A$-bimodule. Let ...
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3answers
80 views

Is it true that $\operatorname{Hom}_R(M,N) \otimes_R P=\operatorname{Hom}_R(M ,N \otimes P)$?

Let $M, N, P$ be modules over a commutative ring $R$. The above identity is true for $R$ a field: Since the RHS $\cong$ (by passing to double dual) $$\operatorname{Hom}(N^* \otimes P^*, ...
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0answers
48 views

Idempotent endomorphisms generate a direct sum decomposition

Show that there is a one-to-one correspondence between indempotents $e\in\operatorname{End}_R(V)$ and direct sum decomposition $V=X\dotplus Y$. Attempt: We have $e^2 = e$. Therefore, we can write an ...
2
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1answer
49 views

Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so.

Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so. In between step is $I^j/I^{j+1}$ is noetherian (artinian) $\forall j$. I ...
1
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1answer
35 views

Example of a non noetherian module $M$ s.t. $M/IM$ is noetherian

What is an example of a non noetherian module $M$ s.t. $M/IM$ is noetherian? What is an example of a non artinian module $M$ s.t. $M/IM$ is artinian? What I was thinking that $k[x_1,...,x_n,...]$ ...
0
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1answer
35 views

For which $n$ is $\mathbb{Z}_2$ a $\mathbb{Z}_n$-module?

Determine the values ​​of $n$ for which $\mathbb{Z}_2$ is a $\mathbb{Z}_n$-module (with action given by "the product $\mathbb{Z}_2$"). If $n=2k+1,$ $k\geq 1$, then: \begin{align*} ...
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1answer
29 views

Confusion regarding finding invariant factors of a matrix.

So I'm having a bit of trouble determining invariant factors of a matrix. Say we have $$ \begin{bmatrix} 2 &0 &0 \\ 0 &9 &0 \\ 0 &0 &6 \end{bmatrix} $$ and I want to find the ...
0
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1answer
18 views

Let $M$ be a noetherian $R$-module and $I=\mathrm{Ann}_R(M)$. Then $R/I$ is a noetherian ring. [duplicate]

Let $M$ be a noetherian $R$-module and $I=\mathrm{Ann}_R(M)$. Then $R/I$ is a noetherian ring. I was trying to show that $R/I$ is a submodule of $M^n$ if $M=\langle f_1,...,f_n\rangle$. Now $M$ ...
1
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1answer
35 views

The structure of a linear endomorphism and its implication to End(V) as a vector space and as an algebra

It is known that a linear endomorphism $T:V\to V$ on an $n$ dimensional $\mathbb{C}$-vector space can be decomposed according to the invariant subspaces of $V$ under $T$. This result can be proved as ...
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1answer
21 views

For A-algebra B,$rank_{A_p}M_p$=$rank_{B_q}M_q$?

Let φ:$A\rightarrow B$ be a A-algebra (commutative ring),and M is a module over B. Now if $M_p$ is a free module over $A_p$ with rank $l$, then $M_q$ is a free module over $B_q$ with same rank $l$? ...
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0answers
33 views

Understanding the Krull-Schmidt-Ramak theorem

I am trying to understand the meaning of so-called Krull-Schmidt theorem, which is stated in terms of both groups and modules. For groups, as I recall, the statement of Krull-Schimidt theorem is as ...
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1answer
85 views

When is $\operatorname{Hom}(M, E)$ injective?

Let $R$ be a commutative Noetherian ring with non-zero identity, $M$ be an $R$-module, and $E$ be an injective $R$-module. When is $\operatorname{Hom}(M, E)$ an injective $R$-module?
3
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1answer
61 views

Why functors that preserve cokernels are right exact?

"Let $F$ be a functor from $A-$modules to $A-$modules that preserves cokernels. Then $F$ take exact short sequences into right exact short sequences." I found a mistake in my proof. I also can't find ...
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0answers
33 views

Does the tensor product commute with the exterior product?

Let $R,S$ be commutative rings, $S$ an extension of $R$ and $A$ a $R$-module. Is it true that $(\bigwedge^{k} A) \otimes_{R} S = \bigwedge^{k} (A \otimes_{R} S)$? I was trying to use the ...
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3answers
42 views

Example needed for $\alpha M=M$, where $M$ is a module, and $\alpha$ an ideal of the ring [closed]

Let $\alpha\lhd R$ be a non-trivial ideal of ring $R$, and let $M$ be an $R$-module. Could someone give me an example where $\alpha M=M$? Motivation: Nakayama's lemma has this as an assumption.
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2answers
51 views

$\mathrm{Tor}(M/\mathrm{Tor}(M))=0$ for an $A$-module $M$?

Definition. Let $M$ be an $A$-module. Define $\mathrm{Tor}(M)=\{m \in M : \exists a \in A-\{0\}, am = 0 \}$). Show that the quotient module $M/\mathrm{Tor}(M)$ doesn't have a non-zero torsion ...
0
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1answer
32 views

Annihilator - Module

I have already shown that if $N$ is a submodule of $M$, then the annihilator $\mathrm{Ann}(N) = \{a \in A : as=0, \forall n\in N \}$ is an ideal of $A$. The other part of the question asks to show ...
1
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1answer
30 views

universal property of the direct colimit

Let $(\Lambda,\le )$ be a directed set, which we can understand as a small category: The set of all objects is $\Lambda$ and for $\lambda,\lambda '\in \Lambda$ there exists an unique morphism ...
3
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1answer
50 views

A matrix with trace $\leq n$ is conjugate to a matrix with all entries $0,1,-1$

Let $A$ be a matrix in $SL(n,\mathbb Z)$, let $a_{ij}$ denote the element in the $i$th row and $j$th column. Suppose $|tr(A)|\leq n$. Is it true that $A$ is conjugate to a matrix $B$ such that ...
0
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1answer
40 views

Are $\mathbb{Z}[x]/(x^2+2)$ and $\mathbb{Z}[\sqrt{-2}] $ isomorphic as Z modules?

Are $\mathbb{Z}[x]/(x^2+2)$ and $\mathbb{Z}[\sqrt{-2}] $ isomorphic as Z modules? I already know they are isomorphic as rings because $x^2+2$ is the minimal polynomial of $\sqrt{-2}$ However, I'm ...
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0answers
14 views

given a surjective ring homoromorphism from R to S, ideals in S are the R submodules of S

A lemma in my lecture notes claims, as in the title, given a surjective ring homoromorphism from R to S, ideals in S are the R submodules of S (in the first paragraph - picture attached), but I can't ...
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1answer
26 views

What are the submodules of $M⊗K$?

Let $k$ be a field and $K$ is a extension of this field. And let $A$ be a finite type $k$-algebra, and $M$ be a finitely generated module over $A$. Then, is the form of submodules of $M⊗_kK$ always ...
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1answer
35 views

Almost Noetherian module

An $R$-module $M$ is called an almost Noetherian $R$-module if every proper submodule of $M$ is finitely generated. My question is the "if" part of (ii): it says that in order to show that $K$ is ...
0
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2answers
62 views

Discrete valuation ring and finitely generated submodules

Let $R$ be a Discrete Valuation Ring with fraction field $K$. Will this imply any proper $R$-submodule of $K$ is finitely generated (hence a fractional ideal)? I know $K$ is not finitely ...
0
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1answer
50 views

Composition factors of injective indecomposable and projective indecomposable modules

Let $A$ be a finite-dimensional algebra over an arbitrary field $K$. Let $L_1$ and $L_2$ be simple modules such that $L_1 \not \cong L_2$. Let the $A$-module $Q_1$ be the injective hull of $L_1$, ...
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2answers
22 views

If N and M/N are free A modules on the ring A with 1, then M is also a free A module

I would like to proove the following : If N and M/N are free A modules on the ring A with 1, then M is also a free A module. My main issue is that we dont know if N and M/N have a finite rank. ...
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0answers
14 views

Show that Soc($\bigoplus_{\alpha\in I}M_\alpha$)=$\bigoplus_{\alpha\in I}\text{Soc}(M_\alpha)$

Given left $R$-modules $\{M_\alpha\}_{\alpha\in I}$ for some index set $I$, I'd like to show that the socle of their direct sum is the direct sum of each module's socle, that is, ...
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0answers
24 views

Let $e \in R$ be an idempotent. Show $Re$ is a projective $R$-module.

Suppose $e$ is an idempotent element of a unital ring $R$. I'm trying to show $Re$ is a projective left $R$-module. I've seen a few answers out there about showing $Re \oplus R(1-e) = R$ to finish the ...
2
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1answer
34 views

Properties of $A-$modules

Let $A$ a ring and $M,N$ two $A-$modules. Let $\varphi:M\longrightarrow N$ a $A-$modules homomorphism. 1) Show that $\ker \varphi$ and $Im(\varphi)$ are respectively submodules of $M$ and $N$. 2) ...
1
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1answer
29 views

Jacobson radical of polynomial quotient ring.

Let $F$ be a field, and $A=F[x]/(x(x-1)^2)$. 1. Find the ideals of $A$. Which of them are simple or maximal? 2. Find the Jacobson radical, $J(A)$, of $A$. 3. Find two composition series for $A$, as ...
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2answers
54 views

Minimal graded free resolution of the ideal $I = (x^r, y^s) \subset k[x,y]$

What is the minimal free-graded resolution of the ideal $I = (x^r, y^s) \subset k[x,y]=R$ for $r,s \in \mathbb{N}$? I tried reducing this down to $r = s = 1$ and I think it is $$0 \to R(-2) \to ...
1
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0answers
13 views

Zero action on a Lie algebra quotient submodule

Let $L $ be a Lie algebra and $I$ be an ideal in $L$. If $M,N$ are $L$-module with $N\subset M$. If $M/N$ is an $L/I$-module and $K/N$ is am $L/I$-submodule of $M/N$ with $(L/I)\cdot (K/N)=0$. Does ...
3
votes
1answer
87 views

Variant of Nakayama's lemma

I am trying to prove that if $M$ is an $R$-module, with $R$ complete w.r.t. an ideal $\mathfrak{m}$, and $M$ is separated ($\cap_k \mathfrak{m}^k M=0$) and the images of $m_1,\dots,m_n$ generate ...
1
vote
1answer
35 views

Existence of injective homomorhism and surjective homomorphism of free modules implies isomorphism [duplicate]

Given an integral domain $R$, we have that $R^n$ embeds in $R^{n+1}$ as free modules over $R$. Can we have a surjective $R$-homomorphism from $R^n$ to $R^{n+1}$? Or more so, given two free $R$-modules ...
1
vote
2answers
40 views

A module over a noetherian ring and its dual

Let $R$ be a noetherian ring, $M$ an $R$-module and $M^*$ be its dual. Is it true that if $M$ is noetherian then $f(x) = 0$ for all $f \in M^*$ implies $x =0$? How about the other way round? Ok, ...
4
votes
1answer
35 views

Apparent Contradiction to Weyl's Theorem

Let $L$ be $sl(2)$, i.e., $L=span\{h,e,f\}$, where $[h,e]=2e$,$[h,f]=-2f$,$[e,f]=h$. This is semi-simple. Suppose I create a module $V=span\{v_1,v_2,v_3\}$ and define actions as follows: ...
1
vote
1answer
34 views

Wedderburn component of $\mathbb C[G]$ corresponding to a contragredient character

Let $G$ be a finite group and let $\phi$ be an irreducible character of $G$ over $\mathbb C$. How does the Wedderburn component of $\mathbb C[G]$ corresponding to the contragredient character of ...
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1answer
33 views

Find all subspaces of the real vector space $\Bbb R^2$.

Find all subspaces of the real vector space $\Bbb R^2$. Is is true for any elements $u=(a, b)$ and $v=(c, d)$ of $\Bbb R^2$,there exists a non-trivial subspace $W$ of $\Bbb R^2$ such that $u, v \, € ...