Questions tagged [module-isomorphism]

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Prove two groups are isomorphic

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
21 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
31 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
46 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
37 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
41 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
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1answer
17 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
17 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 ...
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1answer
57 views

Direct Sum In modules

Definitions: $1)$ Let $M_{\alpha}$, $\alpha \in I$ be a family of submodules of a module $N$ over a Ring $R$. Then we define the sum of the modules $$\sum_{\alpha \in I}M_{\alpha}= \{x_{\alpha _{1}}+...
2
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1answer
24 views

Isomorphism between tensor product of modules and quotient module

I'm trying to show that $M\otimes_{R}R/\mathbf{m}$ and $M/\mathbf{m}M$ are isomorphic as $R$-modules, where $M$ is an arbitrary $R$-Module, $R=k[x_{1},\ldots,x_{n}]$ with a field $k$, $\mathbf{m}=(x_{...
2
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1answer
69 views

Is the Evaluation map of an R-Module of rank 1 and hom injective

This is in context of a larger problem of showing that the dual of an invertible sheaf is invertible on a scheme. I want to show that given a free R-module A of rank 1, the standard evaluation map ...
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0answers
25 views

A problem about checking isomorphism of R-module

Let $R=K\left[x_1,x_2,\cdots,x_n\right]$ be a polynomial ring with coefficients in the field $K$; $\alpha_1,\alpha_2,\cdots,\alpha_p,\beta_1,\beta_2,\cdots,\beta_q\in R^{1\times m}$; $M_1$ be the $R$-...
4
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2answers
202 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 ...
3
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1answer
75 views

Showing that $\frac{\mathbb{R}[x]}{\langle x \rangle}$ and $\frac{\mathbb{R}[x]}{\langle x-1 \rangle}$ are not isomorphic as $\mathbb{R}[x]$ modules.

I'm trying to solve an exercise which asks me to prove that $\frac{\mathbb{R}[x]}{\langle x \rangle}$ and $\frac{\mathbb{R}[x]}{\langle x-1 \rangle}$ are isomorphic as rings, but not as $\mathbb{R}[x]$...
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2answers
114 views

FOL - If two models agree on every sentence are they isomorphic?

Let $M,N$ be two models. If for every sentence $\varphi$, $M\models \varphi \iff N\models \varphi$ then they are isomorphic. My intuition is that that the claim above is incorrect. While the other ...
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0answers
29 views

Isomorphism between simple modules

Let $R$ be a ring let $I$ be a maximal left ideal of $R$. Then we can consider the module $R/I$. This module is simple as $I$ is maximal. Now, take another left ideal $J$ of $R$. If we have an ...
4
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1answer
86 views

Is $R$ finitely generated?

Let $A$ be a commutative ring with identity. Given two submodules $R,S$ of $A^n(n\in\Bbb N)$ and suppose $S$ is finitely generated, if there exists an isomorphism of $A$-modules $A^n/R\simeq A^n/S$, ...
2
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2answers
142 views

Do we have $R\simeq S$ for two submodules $R,S$ of $A^n$?

Let $A$ be a commutative ring with identity. Given two submodules $R,S$ of $A^n$ (where $n\in\Bbb N$), if there exists an isomorphism of $A$-modules $A^n/R\simeq A^n/S$, then do we have $R\simeq S$? ...
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0answers
21 views

Do we have an isomorphism of $A$-modules $R\simeq S$? [duplicate]

Let $A$ be a commutative ring with identity and $M$ an $A$-module. Given two submodules $R,S$ of $M$, if there exists an isomorphism of $A$-modules $M/R\simeq M/S$, then do we have $R\simeq S$?
2
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2answers
44 views

Are the $k[x]$-modules $M = k[x]/\langle x + 1 \rangle \oplus k[x]/\langle x + 1 \rangle $ and $N = k[x]/\langle (x + 1)^2 \rangle$ isomorphic?

Let $k$ be a field and consider the $k[x]$-modules $$M = \frac{k[x]}{\langle x + 1 \rangle} \oplus \frac{k[x]}{\langle x + 1 \rangle} \quad \text{ and } \quad N = \frac{k[x]}{\langle (x + 1)^2 \...
1
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0answers
50 views

Ring, but not a field.

Let $R$ be a ring, $F$ a field, and $\phi:R\rightarrow F$ a ring homomorphism. Suppose that exists a bijection $f:R\rightarrow F$ such that $$f(rx+sy) = \phi(r)f(x)+\phi(s)f(y)$$ for all $r,s,x,y\in R$...
1
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1answer
16 views

$\ell(M/N) \leq \ell(M)$ if $M$ is an $A$-module

$M$ is an $A$-module for some ring $A$. My approach: suppose $\ell(M) < \infty$ because otherwise the proposition is trivial. Let $0 = M_r \subsetneq M_{r-1} \subsetneq ... \subsetneq M_0 = M$ be ...
3
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0answers
85 views

Is the bidual of a C*-algebra isomorphic to the universal enveloping von Nemann algebra as a Banach algebra?

Let $A$ be a C*-algebra and $(\pi, H)$ a universal representation of $A$. We know that there is a linear isomorphism $\tilde{\pi}$ between the bidual $A^{**}$ of $A$ and the universal enveloping von ...
3
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1answer
35 views

Finding an isomorhpism $A_n \cong \mathbb{Z}/(m,n)\mathbb{Z}$

Let me define $A_n = \{ a \in \mathbb{Z}/m\mathbb{Z} : an = 0 \}$. Could anyone give me a hint as to how to construct an explicit bijection between $A_n$ and $\mathbb{Z}/(n,m)\mathbb{Z}$. I really ...
1
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1answer
53 views

Submodules and isomorphisms

Let $S\subset R$ be rings. Denote $R^n$ the free $R-$module of rank $n$, and $S^n$ a $R-$submodule. Let $\alpha, \beta: R^n\rightarrow R^n$ be $R-$module automorphisms such that $\alpha(S^n), \beta(S^...
1
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1answer
159 views

Prove that if A is simple every R-module endomorphism is either the zero map or an isomorphism.

How can I prove that: "Prove that if R has an identity and A is a nonzero unitary R-module and if A is simple every R-module endomorphism is either the zero map or an isomorphism" My answer is: Let $...
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1answer
30 views

Why in the following problem the author take the left ideal in the following form?

This is the question: And this is its answer: Why in the following problem the author take the left ideal in the following form? could anyone explain this for me please?
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1answer
47 views

Show $\phi(B)/\phi(A)\cong B/A\oplus(B\cap\ker\phi)$

I am working on the following problem and keep getting stuck. Let $M$ an $R$-module and $A$ and $B$ submodules with $A\subseteq B$ and $\phi:M\rightarrow N$ an $R$-module homomorphism. Show that: ...
1
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1answer
191 views

The set of all covectors fields and the dual to the set of all vector fields, isomorphic?

Given a smooth manifold $M$, the space of smooth sections of the tangent bundle is denoted by $\Gamma(TM)$ and the space of smooth sections of the cotangent bundle $\Gamma(T^* M)$. Both $\Gamma(TM)$ ...
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0answers
49 views

How to denote isomorphism/ ring direct product

Assume a ring $R$ is isomorphic to direct product of $F_1,F_2,F_3$ We can denote this by $R\cong F_1\times F_2 \times F_3$ Is this equivalent to $R\cong F_1\oplus F_2 \oplus F_3$ ???
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0answers
24 views

Modules where $A+(M/A) \ncong A\oplus M/A$

I'm currently trying to figure out why $\mathbb{C}[x,y] \ncong (x,y) \oplus \mathbb{C}[x,y]/(x,y)$ where $\mathbb{C}[x,y]$ acts on itself by a strange action which I describe below (this action forces ...
1
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1answer
68 views

$M$ be a finitely generated module over commutative unital ring $R$ , $N,P$ submodules , $P\subseteq N \subseteq M$ and $M\cong P$ , is $M\cong N$?

Let $R$ be a commutative ring with unity , $M$ be a finitely generated module over $R$ , let $N,P$ be submodules of $M$ such that $P\subseteq N \subseteq M$ and $M\cong P$ , then is it true that $M\...