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Questions tagged [modules]

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

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1answer
17 views

Why a particular module over a DVR is cyclic if it's quotient and kernel are cyclic?

I'm looking at this lemma below. Here $A$ is a DVR with uniformizer $\pi$. I'm not sure how to justify the underlined step. Why does $M_n$ have to be cyclic if $M_1$ is cyclic?
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15 views

The set of bilinear forms is a right $(R \otimes R)$-module

Let $V$ and $A$ be abelian groups. An $A$-valued bilinear form on $V$ is a $\mathbb{Z}$-module homomorphism $$\beta : V \otimes_{\mathbb{Z}} V \rightarrow A$$ Now, let $V$ be a left $R$-module, where $...
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9 views

Tensor products of modules over non-commutative rings

I've been learning about tensor products over modules, but where the ring acting on the module is commutative. When $R$ is non-commutative, we consider a right $R$-module $M$ and a left $R$-module $N$...
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1answer
22 views

Showing right-exactness of a sequence between modules

I am trying to show the right-exactness of a sequence of a sequence of modules, but I am stuck at the last step. Let $R$ be a ring, and $L, M,N$ be $R$-modules (In the exercise I am trying to solve ...
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0answers
18 views

Proof Verification Short Exact Sequence Rank Theorem

I am trying to prove the rank-nullity theorem for short exact sequences; if $R$ is an integral domain, and $M',\,M,\,M''$ are all $R$-modules with $$0\rightarrow M' \xrightarrow{\psi} M\xrightarrow{\...
1
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1answer
23 views

Isomorphism between quotient rings and modules

Let $I_1, I_2$ be ideals of $R$ — associative ring with unit. Find an example where $R/ I_1$ and $ R/I_2$ isomorphic as rings, but not isomorphic as modules. Can you check my solution? I have an ...
2
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1answer
28 views

$M$ is cyclic if and only if there is some ideal $I\subset R$ such that $R/I\cong M$

This is the first part of exercise 11 on page 168 of Advanced linear algebra third edition of Steven Roman Let $M$ an $R$-module, then prove that $M$ is cyclic if and only if there is some ideal $I\...
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2answers
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Let $R$ be a ring with $1$. If there exist disjoint comaximal ideals $I, J$, then for any $R$-module $M$, $M = IM \oplus JM$.

This questions is from a past year paper. Let $R$ be a ring with $1$. Suppose there exist distinct ideals $I, J$ of $R$ such that $R = I + J$ and $I \cap J = \{ 0 \}$. (a) Show that there exists $a \...
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0answers
37 views

Modules, endomorphism and ideals

I was solving tasks in a book and there are two tasks where I have no idea how to solve them: 1) Let $R$ be commutative ring, $f: M\rightarrow N$ - isomorphism of $R$-modules. Prove isomorphism of ...
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1answer
39 views

Torsion Free Module over Dedeking Ring

Let $\phi: R \to A$ be a finite morphism of Dedekind rings (so $A$ is a finitely generated $R$-module) and $M$ a finitely generated $A$-module. Obviously, if we restrict the action of $A$ on $M$ to $R$...
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27 views

Quotient of Module over PID

Let $R$ be a PID Let $a$ be a nonzero element in $R$ Let $M=R/(a)$ For any p of R prove that $p^{k-1}M/p^kM\cong R/(p)$ if $k\leq n$ $p^{k-1}M/p^kM\cong 0$ if $k> n$ where $n$ is the power ...
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58 views

$R^{\mathbb N}$ as a free $R$-module.

Suppose that $R$ is a commutative ring. I'm wondering if the space $R^{\mathbb N}$ is a free $R$ module. I know how to prove that it is not a free $R$ module in the case of $R = \mathbb Z$. But the ...
2
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2answers
62 views

Prove that any $R$-module $M$ is isomorphic to $\mathrm{hom}_R(R,M)$

This is an exercise of Advanced linear algebra third edition of Steven Roman: Prove that any $R$-module $M$ is isomorphic to $\mathrm{hom}_R(R,M)$ My work so far: We want to show that $M\approx\...
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22 views

Equivalence of module structure involving an $A$-algebra

I'm new to modules and I'm trying to see the following equivalence: "If $A$ is a commutative ring, $R$ an $A$-algebra (i.e. a ring $R$ with a homomorphism $i : A \rightarrow Z(R)$) and $M$ an Abelian ...
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Intersection with direct sum of modules [closed]

Let $R$ be an arbitrary ring and $M=\bigoplus_{i\in I} M_{i}$ be a direct sum of $R$-modules. Give an example shows that for a submodule $N$ of $M$ it is not necessary that $N=\bigoplus_{i\in I}N\...
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1answer
29 views

Tensor product of module homomorphisms is element of tensor product or mapping?

Here: $A$ is a ring. $E,E'$ are right $A$-modules, $F,F'$ are left $A$-modules. $u:E\rightarrow E'$ and $v:F\rightarrow F'$ are $A$-module homomorphisms. With this setup, I didn't understand ...
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1answer
30 views

Tensor product of modules: should it be abelian group or module?

If $A$ is a right $R$-module and $B$ is a left $R$-module, then the tensor product $A\otimes_R B$ is an object $X$ with a map $\theta:A\times B\rightarrow X$ such that $\theta$ is $R$-bilinear and $\...
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0answers
12 views

Loewy length of a submodule

Let M be a module with Loewy length k and N be a submodule of M. Is the Loewy length of N $\leq$ the Loewy length of M?
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1answer
13 views

To find annihilator of given module

I want to find annihilator of $Z_{14}$ and $Z_4 × Z_6$ as Z module. So in first case element a from Z will be in annihilator of $Z_{14}$ if 14 divides a,2a,3a,....,13a. But from here I am not getting ...
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1answer
17 views

The flat module, module is not flat

Why $\mathbb{Z}$-module $\mathbb{Q}$ is flat and $\mathbb{Z}$-module $\mathbb{Z}_n$ is not flat? P/s: How can I prove them by definition and without functor. Thankyou.
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1answer
22 views

Baer sum, pushback of pushout and pushout of pullback

Let us consider the following constructions in the category of $R-$modules, for some ring $R$. Given a short exact sequence $$ \mathcal{S}: \quad 0 \to A \overset{\alpha}{\to} B \overset{\beta}{\to} ...
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1answer
28 views

When do tensor products of elements coincide

Let $M,N$ be $R$-modules and $m \otimes n, m' \otimes n' \in M \otimes_R N$ non-zero (EDIT) elements. When does $m \otimes n = m' \otimes n'$ hold? Obviously, this is true if either $(m',n)=(rm,rn')$ ...
4
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2answers
193 views

Are two submodules (where one is contained in the other) isomorphic if their quotientmodules are isomorphic?

Let $M$ be an $R$ module and $N_1 \subset N_2$ be submodules of $M$ such that $M / N_1 \cong M / N_2$. Can I know conclude $N_1 \cong N_2$ or even $N_1 = N_2$? I know that a proper submodule can be ...
2
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2answers
38 views

Identifying simple tensors.

Let $S$ be a domain. I want to determine whether or not, every element of $\text{Frac(S)}\otimes_S M$ is a simple tensor, where $M$ is any $S$-module. I couldn't produce a tensor that is not pure in ...
2
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1answer
28 views

Nonsurjective map to second dual module

For a free finitely generated module $M$ the canonical map $i: M\to M^{**}$ is an isomorphism. I know an example of a finitely generated module for which this map is not injective. The map is also not ...
1
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1answer
24 views

Is $X\cong\textrm{Ker}(f)\oplus\textrm{Im}(f)$ for a module homomorphism $f:X\to Y$ with semisimple domain?

Let $f:X\to Y$ be a module homomorphism with semisimple domain. Does \begin{equation*} X\cong\textrm{Ker}(f)\oplus\textrm{Im}(f) \end{equation*} hold true that? (In my previous question it was kindly ...
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1answer
52 views

Extension of basis over PID [closed]

Let $k[x] = R$ be ring and $L$ be free $k[x]$-module; let $v \in L$ be vector in $L$. Then how one can extend it to an $R$-basis for $L$?
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2answers
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Proof $F$ is a free module on $X$

Let $\{X_i:i\in I\}$ be a collection of mutually disjoint sets and for each $i \in I$ let $F_i$ be a free module on $X_i$ with $l_i: X_i\rightarrow F_i$. Let $X=\bigcup_{i\in I}X_i$ and $F=\sum_{i\in ...
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1answer
26 views

Free module definition

Let $X$ be non empty set and let $F$ be a left $R$ module. Left $R$ module with function $f:X\rightarrow A$ is called free on $X$ if exists unique homomorphism of $R$ modules $g: F\rightarrow A$ such ...
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1answer
70 views

Annihilator of a finitely generated torsion module is nonzero? [duplicate]

Let $R$ be a commutative ring and $M$ a finitely generated torsion $R$-module. Then the ideal $\mathrm{Ann}(M)$ is nonzero? If $R$ is integral domain, this trivially holds since any two ideals have ...
5
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1answer
93 views

$\operatorname{Frac}(A)/A$ as an $A$-module

I am wondering about a question: We know that $\mathbb{Q}/\mathbb{Z}$ is torsion group and $\mathbb{Q}/\mathbb{Z}=\bigoplus_{p\text{ prime}}\mathbb{Q}/\mathbb{Z}(p)$ where $\mathbb{Q}/\mathbb{Z}(p)=\{...
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1answer
23 views

Hilbert Syzygy Theorem for non-graded modules

The statement of Hilbert Syzygy Theorem is as follows: Let $R = k[x_1 , \ldots , x_n]$ be a polynomial ring over a field $k$ and $M$ be a finitely generated graded $R$-module. Then $\text{pd }M \leq n$...
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52 views

Does $X/K\cong Y/K$ imply $X\cong Y$ for modules?

Let $X$ and $Y$ be left modules over a ring $R$, and let $K$ be a submodule of both $X$ and $Y$ such that $X/K\cong Y/K$. Does this imply that $X\cong Y$? In the particular case that $X$ and $Y$ are ...
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29 views

For an equivalent functor $\Sigma:T \to T$, if $T(v \circ u)=0$ then $v \circ u=0$.

Im beginnig selfstudy of triangulated categories, and Im working with an additive category $T$ and an additive covariant and equivalent functor $\Sigma:T \to T$, as equivalent the notes say that ...
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1answer
25 views

Decomposition of $K[X]$-modules, where $K$ a field

I saw something like "$K[X]/(X^2) \cong K[X]/(X) \oplus K[X]/(X)$ is not true because otherwise $K[X]/(X^2)$ would be annihilated by the action of X". What does it mean exactly? Also, I saw $R^2 \...
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0answers
17 views

Decomposition of $C[G]$ as $C[G]$-module into direct sum of submodules of the form $Ce_{\chi}$

Let C be the complex field, and $G$ be a cyclic group generated by $a$. The group ring $C[G]$ is a $C[G]$-module (over itself) with the module action $C[G]\times C[G] \rightarrow C[G]$ the same as the ...
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1answer
46 views

Some basic isomorphisms in rings and modules

Let $R$ be a ring with $1$. Let $M$ be a left $R$-module and $e$ an idempotent in $R$. Then (1) ${\rm Hom}_R(R,M)\cong M$ (isomorphic abelian groups). (2) ${\rm Hom}_R(Re,M)\cong eM$ (isomorphic ...
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23 views

Every finite dimensional algebra is subalgebra of a matrix algebra

I have two questions about the following exercise: Let $K$ be a field and $R$ a $K$-algebra. Assume that $d := dim_K R$ is finite. Show that there is an injective $K$-algebra homomorphism $...
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1answer
25 views

What does this Module notation mean?

I have to prove that if $\{x_i\}_{i\in I}$ is a base of a A-module M, then: $$M=\underset{i\in I}{\oplus}Ax_i=\underset{i\in I}{\oplus}(x_i)$$ Am I right to assume that I have to prove that $\sum_{i\...
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1answer
41 views

Decomposability of a module

I study by myself algebra first undergraduate course. I follow the book (https://www.springer.com/us/book/9783319452845#reviews). I am trying to understand decomposability of modules (chapter 14). I ...
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0answers
45 views

$K[X]$ acts on $V$ by linear transformation $T$

Let $K$ be a field, $n\geq 1$ and $V$ a $K$-vector space of dimension $n$. Let $T: V \to V$ be a linear map, I'm not sure how to show that the ideal of all polynomials $P \in K[X]$ such that $P(T) = 0$...
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0answers
33 views

Quotient of quotients in finite dimensional vector spaces

Suppose we are given filtrations of finite dimensional vectors spaces: $$ B_d\subseteq Z_d\subseteq C_d$$ $$0 \subseteq C_{d,1} \subseteq C_d$$ $$0 \subseteq Z_{d,1} \subseteq Z_d$$ $$0 \subseteq B_{...
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0answers
35 views

Betti numbers of finitely generated module over Noetherian local ring, after going modulo a regular element

For a finitely generated module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ , let $b_i^R(M):= \dim_k \operatorname{Tor}_i^R (k,M)$. It is known that this $i$-th Betti numbers ...
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1answer
34 views

Definition of Internal Direct Sum of Modules

Here is the definition of an internal direct sum of modules: An R-Module $M$ is the internal direct sum of submodules $M_1, M_2$ if: $a)$ $M=M_1+M_2$ $b)$ $M_1 \cap M_2 = \{0\}$ I am ...
1
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1answer
20 views

Question about a near $\Bbb{Z}$-semimodule that has trivial addition.

https://en.wikipedia.org/wiki/Semimodule Suppose that we have a structure $M$ that has: Closure under addition. Closure under multiplication by any $a \in\Bbb{Z}$. The addition of any two elements ...
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1answer
28 views

Problem reagarding the annihilator of a $\mathbb Z$-module

Let $M=\mathbb Z/24\mathbb Z\times \mathbb Z/15\mathbb Z\times \mathbb Z/50\mathbb Z$. a) Find the annihilator of $M$ in $\mathbb Z$. b) Let $I=2\mathbb Z$. Describe the annihilator of $I$ in $M$ ...
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1answer
44 views

For an integral extension $A \subseteq B$, $\sqrt{IB}\cap A=\sqrt{I}$

Let $A \subseteq B$ be an integral extension and $I \subseteq A$ an ideal. Prove that $\sqrt{IB}\cap A=\sqrt{I}$. I saw a similar property holds for Jacobson radical where the following two facts ...
2
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2answers
48 views

Prove that the map $f: \Bbb C \times \Bbb C \to \Bbb C \times \Bbb C$ by $f(z_1,z_2)=(z_1z_2,z_1\bar{z_2})$ is an $\Bbb R $ bilinear map

I am trying to solve the question 27 of Section 10.4 of Dummit and Foote but I am stuck in the first problem: let me state the question and then I will attach the picture of the page of the ...
0
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1answer
38 views

Every short exact sequence with a simple module splits

More concretely, if $R$ is a ring, and $M$ is a simple $R$-module, I want to show that any short exact sequence $0 \rightarrow L \rightarrow M \rightarrow N \rightarrow0$ splits. To this end, I have ...