Questions tagged [module-isomorphism]

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Let $M_1$ and $M_2$ be $R$-modules, show that $N_1×N_2$ is a submodule of $M_1×M_2$

Two parts of a class exercise. Let $M_1$ and $M_2$ be $R$-modules, show that $N_1×N_2$ is a submodule of $M_1×M_2$, given that $N_1$ and $N_2$ are submodules of $M_1$ and $M_2$, respectively. Show ...
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36 views

linearly isomorphic submodule

I'm currently learning about modules and am pretty new to the topic. I recently found this question: Suppose that $R$ is an integral domain and not a field. Give, with proof, an eample of a proper ...
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module isomorphisms

so I am currently learning about modules, I'm pretty new to them, but i have some experience with rings and linear algebra and stuff. I got the following problem: Let $R$ be the ring $\mathbb{Z}[\...
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1answer
75 views

Understanding example 3 on pg. 369 in Dummit & Foote (3rd edition)

Here is the example: $(3)$ In general, $$ \mathbb Z_m \otimes_Z \mathbb Z_n \cong \mathbb Z/d\mathbb Z,$$where $d$ is the $g.c.d$ of the integers $m$ and $n.$ To see this, observe first that $$a \...
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Proving $r(m + n) = rm + rn$ for a type of scalar multiplication.

Here is the question I want to answer: Let $R \subset S$ be commutative rings and let $M$ be an $R$-module. Then $S \otimes_R M$ is an $R$-module generated by $\{s \otimes m \mid s \in S, m \in M\}.$ ...
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1answer
45 views

Understanding how the hint will prove injectivity.

Here is the question I want to solve: Let $M$ be a finitely generated $R$-module. Show that if $f \in \mathrm{End}_R(M)$ is surjective then it is also injective. And here is the hint I got for the ...
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47 views

Proving that $M$ is a simple $R-$module.

Here is the question I want to answer: A module is simple if it is not the zero module and it has no proper nonzero submodule. $(a)$ Let $M$ be an $R-$module. Show that the following conditions are ...
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44 views

Do I have to show that elements of $L$ commutes with elements of $N$ (like in case of direct product) and if so, why?

I want to prove the following $a \Longleftrightarrow d$ in the following questoin: Let $R$ be a commutative ring. For $R-$modules $L,M,N$ show that the following conditions are equivalent.(all ...
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1answer
49 views

Proving that the existence of a section implies the direct sum.

Here is the question I want to prove so far: Let $R$ be a commutative ring. For $R-$modules $L,M,N$ show that the following conditions are equivalent.(all functions are $R-$ module homomorphisms.) a- $...
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1answer
31 views

Module Isomorphism

Let X, Y , and Z be $R$- Modules, for $R$ a commutative ring. Furthermore, let $X$ and $Y$ be submodules of $Z$. I want to show that if $X + Y = Z$ and $X \cap Y = \{0 \}$, then $Z \cong X \oplus Y$. ...
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1answer
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Finding Tensor Product [duplicate]

Let $\Pi_{n\in \mathbb{N}}\mathbb{Z}:= M$ Is $ M \otimes_{\mathbb{Z}} \mathbb{Q} \cong \Pi_{n\in \mathbb{N}}\mathbb{Q}$ ? I believe this is true but I don't know how to prove this. Kindly help me with ...
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38 views

A problem on module homomorphism

Let $R$ be a commutative ring. Prove that $\text{Hom}_R(R,M)$ and $M$ are isomorphic as left $R$-mod. Question Let $R$ be a commutative ring. Prove that $\text{Hom}_R(R,M)$ and $M$ are isomorphic as ...
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89 views

prove that $\operatorname{Hom}_\Bbb{Z}(\Bbb{Z}/n\Bbb{Z},A)\cong A_n.$

Let $A$ be any $\Bbb{Z}$-module, let $a$ be any element of $A$ and let $n\in \Bbb{Z}^+$. Prove that the map $\psi_a\colon\Bbb{Z/nZ}\to A$ given by $\psi_a(\overline{k})=ka$ is a well-defined $\Bbb{Z}$-...
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56 views

Isomorphism between $R = \mathbb{R}[X, Y]/(X^2 + Y^2 - 1)$ and $IJ$ where $I = (x - 1, y)$ and $J = (y, x - 1)$

Let $R = \mathbb{R}[X, Y]/(X^2 + Y^2 - 1)$ and $I = (x - 1, y)$, $J = (x, y - 1)$, where $x = X + (X^2 + Y^2 - 1)$ and $y = Y + (X^2 + Y^2 - 1)$. I have managed to prove that $I + J = R$ and $IJ = (x +...
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$\mathcal{L}$ embedding | Trasnfering embedding to a new structure

My you help me with this task: Let $\{\mathcal{M}_{n}:n \in \mathbb{N}\}$ a collection of $\mathcal{L}-$structures such that for all $n \in \mathbb{N}$ exists a $\mathcal{L}-$embedding $f_{n}: \...
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1answer
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Idempotent Endomorphism of semi simple module

I try to show the following exercise from Group Representation Theory Book from Peter Webb: Let $A$ be a ring with $1$. Let $V$ be a semisimple $A$-module with finitely many simple summands and $e,f \...
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Module Isomorphisms of Maximal Orders

I have been reading the paper Steinitz classes of central simple algebras and they make the following claim, just above Corollary 3.2: Let $K$ be a number field with ring of integers $\mathcal{O}_K$. ...
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Relationship between submodules of $F[x]/(x^n)$ and $F^n$.

I am working on the following question: Let $F$ be a field. Let $F^n$ be an $F[x]$ module where $x$ acts on $(a_1, ..., a_n) \in F^n$ by $x \cdot (a_1, \dots, a_n) = (0, a_1, \dots, a_{n - 1})$. ...
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For a ring R, CI-rad(M)=rad(M) for each R-module M iff every co-isosimple R-module is simple.

Definition: A non-zero module M is co-isosimple if it is isomorphic to all its nonzero quotients. Definition: A proper submodule N of a non-zero module M is isomaximal if M/N is a co-isosimple module. ...
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Condition under which a ring is isomorphic to a right ideal.

Defintion: an element $a\in R$ is said to be right regular if, there is no nonzero element $b\in R$ such that $ab=0$. My Questuion: Let $R$ be a ring with identity. Then $R$ is $R$-isomorphic to a ...
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Lattice Homomorphisms and Modules

Suppose $M$ and $N$ are finitely generated modules over some (possibly non-commutative) ring $R$. Let there be a lattice isomorphism between the lattice of submodules of $M$ and those of $N$ (i.e. ...
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1answer
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If a left module M is co-isosimple and semi-Hopfian, then M is simple.

I have tried to prove following theorem; The followings are equivalent for a left module M: M is simple, M is co-isosimple and co-regular, M is co-isosimple and semi-Hopfian, M is co-isosimple and ...
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2answers
52 views

$\operatorname{Hom}_R(F,R)$ is isomorphic to $F$ where $F$ is a free $R$-module of finite rank

[Dummit and Foote, Exercises 10.3, problem 13] Suppose $R$ is a commutative ring with $1$. Suppose $F$ is a free $R$-module of finite rank. Then show that $\operatorname{Hom}_R(F,R) \cong F$. Suppose ...
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Quotients of Two-sided and One-sided Ideals

Let $\mathcal{A}$ be a central simple algebra over an algebraic number field $K$, and $\mathcal{O}$ be a maximal $\mathcal{O}_K$-order in $\mathcal{A}$. Let $I$ be a maximal integral left-ideal of $\...
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Proving that $0 \rightarrow \Bbb Z \rightarrow \Bbb Q \rightarrow \Bbb Q / \Bbb Z \rightarrow 0$ does not split.

Can I prove that the $\Bbb Z$-module exact sequence $0 \rightarrow \Bbb Z \rightarrow \Bbb Q \rightarrow \Bbb Q / \Bbb Z \rightarrow 0$ is non-split exact by proving that $\Bbb{Z} \bigoplus \Bbb{Q} / \...
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If $f_x$ is injective whenever it is surjective on $M$, then is $f_x$ injective whenever it is surjective on $\bigoplus_{i\in \Bbb N}M$?

Let $R$ be a commutative ring with unity. For any $x\in R$, let the endomorphism of $M$ induced by multiplication by $x$ be $f_x$, ie., $f_x(m)=mx$. Prove the following: (1) If $f_x$ is surjective ...
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1answer
34 views

$R$-module homomorphism

Let be $R$ a ring, $M$ a $R$ -module, $M_{1}$ and $M_{2}$ two submodules of $M$ and $\phi: M_{1} \oplus M_{2} \rightarrow M$ homomorphism defined by $\phi\left(x_{1}, x_{2}\right)=x_{1}+x_{2}$ then $\...
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Why $M/\mathfrak{a}M \oplus M/\mathfrak{b}M \simeq M/(\mathfrak{a \cap b})M$?

Let $M$ be an $A$-module and let $\mathfrak{a}$ and $\mathfrak{b}$ be coprime ideals of A. I must show that $M/ \mathfrak{a}M \oplus M/ \mathfrak{b}M \simeq M/ (\mathfrak{a \cap b})M$. My attempt is ...
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1answer
108 views

A module homomorphism is injective iff the kernel is isomorphic to the trivial module

Consider a $R$-module homomorphism $\varphi\colon M\to N$. It is well-known that $\ker\varphi=\{0\}$ iff $\varphi$ is injective (equivalently is monic). However, suppose we only have $\ker\varphi\...
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1answer
76 views

Correspondence Theorem for Lie algebras

I am trying to prove the following: Let $L$ be a Lie algebra and $I$ an Ideal of $L$. There exists a bijection between the Ideals of the quotient algebra $L/I$ and the Ideals of $L$, that contain $I$...
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$Hom_R(R,R/I)$ isomorphic to R/I as R-modules

If R is a ring and $I \subset R$ is an ideal. How can we show that $Hom_R(R,R/I)$ isomorphic to R/I as R-modules? Do we need to choose an f in our $Hom_R(R,R/I)$ and show that we have a bijective ...
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Example of a non-isomorphic map from $D^{-1}R$-modules [duplicate]

Let's suppose $R$ is a ring and $D \subset R$ is a multiplicative subset. If we look at these two $D^{-1}R$-modules: $D^{-1}(\operatorname{Hom}_R(M,N))$ and $\operatorname{Hom}_{D^{-1}R}(D^{-1}M,D^...
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Does the isomorphism $\text{Ann}^M(1-ab)\cong \left(\text{Ann}^M(1-ab)\right)a$ always hold?

Let $R$ be a ring with unity and $a,b\in R$. For a right $R$ module $M$, define $\text{Ann}^M(r)=\{m\in M:mr=0\}$ as a right $R$-module: My question is, does the isomorphism $\text{Ann}^M(1-ab)...
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1answer
38 views

Colon ideal and Cyclic modules

I'm reading module theory as a beginner. The following problem might be super silly. I apologize for flooding SE with this kind of basic question. Let $R$ be a ring with $1$, and $\mathscr{m}$ and $\...
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62 views

surjective homomorphism between free abelian groups and first isomorphism theorem

So i recently started studying homology groups and whenever i stumbled upon its general idea or motivation, i came across computations such as $$ \langle a,b,c\rangle / \langle c \rangle = \langle a,...
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Describe all $R$-modules of low order (up to isomorphism) [duplicate]

In old exams, I've encountered the two following similar exercises: (1) Classify, up to isomorphism, all the unitary $\mathbb{Z}[i]$-modules with 10 elements. (2) Classify, up to isomorphism, all ...
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Showing a matrix is 1-1

How do I show below is 1-1. I also know the natural basis is $R^2$
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1answer
54 views

Proof of the isomorphism $A\otimes_{K} M_{n}(K)\cong M_{n}(A)$

Let $A$ be a $K-$algebra. I want to prove that $A\otimes_{K} M_{n}(K)\cong M_{n}(A)$, where $M_{n}$ are all the $n\times n$ matrices over $K$. If we define $f:A\times M_{n}(K) \to M_{n}(A)$ which maps ...
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1answer
29 views

A problem about isomorphism and submodules [closed]

R is a ring with 1, and M, N two modules over R . Suppose that M is isomorphic to some submodule of N, and N is isomorphic to some submodule of M, then is M isomorphic to N?
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1answer
31 views

Prove the modules isomorphism theorem

$(\Rightarrow)$ Given $V_1\cong V_2$, there exist an isomorphism $\gamma:V_1\to V_2$. Since $\alpha_1\in \operatorname{End}_k(V_1)$ and $\alpha_2\in \operatorname{End}_k(V_2)$, $\alpha_1:V_1\to V_1$...
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1answer
17 views

Proving certain map is a k-linear isomorphism between $M^*\otimes N$ and $Mod_k(M,N)$

In Eiichi Abe Hopf Algebras volume we find the following exercice: Given a field $k$, let $M,N$ be $k$-vector spaces with dual $k$-vector spaces $M^*$, $N^*$ respectively. Define a map $\varphi:M^*\...
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179 views

Prove the direct sum of modules is isomorphic

Suppose that $M_1$, $M_2$, $N_1$, $N_2$ are $R$-module and $M_1\cong N_1$, $M_2\cong N_2$. Prove that $M_1\oplus M_2\cong N_1\oplus N_2$. I know that $M_1\cong N_1$ imply there exist an isomorphism $...
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1answer
85 views

Prove two groups are isomorphic [closed]

I have created the two tables but can not find a one to one correlation between the values in the two tables. I would appreciate it if anyone can point me in the right direction to understand how to ...
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1answer
26 views

Question about isomorphism of direct sums (Step needed to show uniqueness of Elementary Divisors Theorem)

I am trying to proof the uniqueness of Elementary Divisors Theorem, I am stuck in this step: Let $A$ be a principal domain and $M_1, M_2$ two finitely generated $A-$modules. Suppose that $$A^n \...
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1answer
59 views

Krull dimension of a module and isomorphisms

I've been working with the Krull dimension of an $R$-module $M$ defined as the deviation of the lattice of submodules of $M$, i.e. $\operatorname{Kdim}(M)=\operatorname{dev}(\delta(M))$. I have been ...
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1answer
192 views

Short Split Exact Sequence Theorem

Let $E:0\rightarrow A\xrightarrow{i} B\xrightarrow{q} C \rightarrow 0$ be a short exact sequence of $R$-modules. Then the following are equivalent : $(1)$ there is an $R$-module homomorphism $\...
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3answers
740 views

Quotient Modules and their Direct Sum

Let $R$ be a ring and $I$ be an ideal of $R$. Let $M$ be a module over $R$.Let $N_{1}$, $N_{2}$ be submodules of $M$ such that M = $N_{1}$ $\oplus$ $N_{2}$. Show that $a)$ $M/IM$ is a module over ...
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2answers
58 views

Fintiely Generated Modules Over PID

Let $M$ be a finitely generated module over a principal Ideal domain $R$. Show that there exists an $R$-module homomorphism $f: R^m\rightarrow R^n$ such that $M\simeq \frac{R^n}{\operatorname{Im}(...
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1answer
93 views

Direct Sum Of Torsion modules and Proof Verification

Let $M$ be a (left) module over an integral domain $R$ and $N_{1}$, $N_{2}$ be submodules of $M$ such that $M = N_{1}\oplus N_{2}$. Let $\operatorname{Tor}(M)$ denote the torsion module of $M$. Then ...
2
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0answers
66 views

Direct Sum of Sub modules and proof Verification

Let $M$ be an $R$-module and $N_{1},N_{2}$ be submodules of $M$ such that $M = N_{1}\oplus N_{2}$. Then $M/N_{1}\simeq N_{2}$ and $M/N_{2}\simeq N_{1}$. My Attempt: Define a $R$-module ...