Questions tagged [module-isomorphism]
The module-isomorphism tag has no usage guidance.
72
questions
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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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2answers
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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0answers
40 views
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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1answer
51 views
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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1answer
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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1answer
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
28 views
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 ...
0
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1answer
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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2answers
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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1answer
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}: \...
2
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1answer
44 views
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 \...
2
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0answers
25 views
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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22 views
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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0answers
17 views
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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1answer
43 views
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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0answers
23 views
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. ...
2
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1answer
36 views
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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0answers
81 views
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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1answer
90 views
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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1answer
27 views
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 ...
1
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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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3answers
82 views
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 ...
1
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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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1answer
30 views
$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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1answer
87 views
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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0answers
47 views
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)...
1
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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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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,...
2
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0answers
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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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0answers
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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 ...
0
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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?
1
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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 $\alpha_2\in \operatorname{End}_k(V_2)$, $\alpha_1:V_1\to V_1$...
0
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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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1answer
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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votes
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 ...
0
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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 \...
0
votes
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 ...
1
vote
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 $\...
0
votes
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
...
0
votes
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}(...
2
votes
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 ...
