Questions tagged [module-isomorphism]

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Chinese Remainder Theorem in Structure Theorem for f.g. Modules over a PID

I have been asked by my Algebra professor, to explicitly determine the Chinese Remainder Theorem in the proof of the Structure Theorem for f.g. Modules over a PID. Here's what I know: $\textbf{...
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Does localisation (of modules) cancel.

If I have two modules $R$ and $M$ and I take the localisation of both then do we have, $$\frac{S^{-1}R}{S^{-1}M}\cong\frac{R}{M}$$
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If $R$ is commutative then $\operatorname{Hom}_R(R,M)$ is isomorphic with $R$-module $M$. Is the converse also true?

I'm reading Elements of Abstract and Linear Algebra by E.H. Connell and I'm wondering if the converse of the following theorem is true (p.71): Theorem Suppose $M=M_R$ and $f,g: R\to M$ are ...
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Condition for isomorphism of $R$-module homomorphism.

In the proof of theorem 9.4.7 of Charles Weibel's "An Introduction to Homological Algebra" it is claimed to aid the proof that an $R$-module homomorphism $\psi$ is an "isomorphism if ...
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Why is $(R/J)/I(R/J)$ isomorphic to $R/(I+J)$ as $R$-modules? [duplicate]

Let $R$ be a commutative ring and let $I, J$ be ideals of $R$. Our professor mentioned in algebra class that the $R$-module isomorphism $R/I \otimes_R R/J \cong R/(I + J)$ can be directly proved from ...
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Prove $\theta:S\to E^G$ is $R$-module isomorphism

I'm currently studying the paper Galois Theory and Galois Cohomology of Commutative Rings by S.U. Chase, D.K. Harrison and Alex Rosenberg. I'm trying to understand step by step the proof of Theorem 1....
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Reverse direction of cyclic $R$-module implies isomorphism

I have some difficulties understanding the answer to this question, Isomorphism and cyclic modules. You might also assume that we are working with commutative unital rings. If we have $M \cong R / I$, ...
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$\mathbb{Z}[i]$-modules with 101 elements and cyclic $\mathbb{Z}[i]$-torsion modules [duplicate]

Well I am solving old exam tests and I am confused. It is asking to find two non-isomorfic $\mathbb{Z}[i]$-modules with 101 elements each, but I think this can't happen because as abelian groups both ...
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Show that two modules $V_A$, $V_B$ are isomorphic

Let $K$ be a field and define two matrices: \begin{align*} A:= \begin{pmatrix} 1&1\\ 0&3\end{pmatrix} \text{ and } B:= \begin{pmatrix} 1&0\\ 0&3\end{pmatrix} \end{align*} We then ...
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Over a left V-ring, why is every module containing a simple essential submodule is simple?

I've tried to prove the statement $R$ is a $V-$ring if and only if every module containing a simple essential submodule is simple. I know that a ring $R$ is called as a $V-$ring if every simple $R$ ...
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Every left $KG$-module $M$ is a right $KG$-module

I don't know how to prove that a left $KG$-module is a right $KG$-module. What I have so far is that a left $G$-module is a right $G$-module always defining the operation $x\cdot'g=g^{-1}\cdot x$. But ...
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Doubts regarding correspondence theorem.

I am having a confusion regarding the correspondence theorem for rings.I want to know if the following is correct. Let $R$ be a ring and $I$ be an ideal of $R$.Consider the sets $\mathcal G$ and $\...
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Proof for $P(R) \leq I-rad(R_R)$

The (right) $\textbf{isoradical}$ $I-rad(R_R)$ of a ring $R$ is the intersection of the annihilators of all isosimple right $R-$modules where an $\textbf{isosimple}$ module is defined as a non-zero ...
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Completely virtually semisimple modules are direct sums of isosimple modules.

An $\textbf{isosimple}$ module is defined as a non-zero module whose all non­zero submodules are isomorphic to it. An $R$-module $M$ is called $\textbf{virtually semisimple}$ if every submodule of $M$ ...
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Some Characterizations of Isoartinian Module

Let $R$ be a ring and $M$ be a right $R$-module. $M$ is called $\textbf{isoartinian}$ if, for every descending chain $M \geq M_1 \geq M_2 \cdots$ of submodules of $M$, there can be found an index $n$ $...
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Equality of submodules satisfying given conditions - Verification of solution

Question: Let $M$ be a left $R$-module($R$ is a unitary ring). let $M_1,M_2,N$ be $R-$submodules of $M$ such that $M_1\subseteq M_2$, $M_1+N=M_2+N$ and $M_1 \cap N=M_2 \cap N$. Are $M_1,M_2$ ...
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Proof verification of showing that $M\otimes A/{\frak a} \cong M/{\frak a}M$

Let $A$ be a (commutative) Ring with $1$. Let $M$ be an $A$-Module and ${\frak a}\subseteq A$ an Ideal. I was trying to prove that $M\otimes A/{\frak a} \cong M/{\frak a}M$. I would like to know ...
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A semiprime right semihereditary ring R is right isonoetherian or right uniform if and only if it is right nei-Noetherian.

I need to prove this theorem, but I guess I need some hints since I cannot even imagine how it can be proved. By the way, it is known that if $R$ is isonoetherian, then it is nei-Noetherian. The ...
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Show that following modules are isomorphic

I have $M=\mathbb{Z^2/\langle(-2,5)\rangle}$ as $\mathbb{Z}$-module and have to show that it is isomorphic with $\mathbb{Z}$. I have found an isomorphism $f: \mathbb{Z^2}\rightarrow\mathbb{Z^2}$ ...
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Hom( , ) and Hom_F( , ) and tensor product

My professor said in his lecture, "For Abelian groups $A$, $B$, $C$ and field $F$ $\operatorname{Hom}(A,B)=\operatorname{Hom}_\mathbb{Z} (A, B)$ but $\operatorname{Hom}(C, F) \neq \operatorname{...
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Prove that the following quotients are isomorphic

I am preparing myself for an upcoming exam, and I've found the following problem Let $M$ be a $R$-module and $N_1 \subset N_2 \subset M$ be $A$-submodules. Use the Snake Lemma to show that $$\frac{M/...
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Submodule of a noetherian module is noetherian [duplicate]

Assume $R$ is a ring, $\psi:M\rightarrow N$ a surjective $R$-modules homomorphism. Then, $M$ is noetherian $\iff$ $N$ and $K=\operatorname{Ker}(\psi)$ are noetherian. So I proved $\leftarrow$. Now I ...
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Does taking covariant Hom commute with taking (co-)kernel?

Let $A$ be a Commutative ring and $R$ be a commutative $A$-algebra. So every $R$-module has a natural $A$-module structure, and for every $A$-module $W$, and $R$-module $M$, the $A$-module $\text{Hom}...
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A co-isosimple module is co-cyclic.

I have tried to prove the statement that if an R-module is co-isosimple, then it is co-cyclic. In the paper that this stated, this is mentioned as since a co-isosimple module is artinian and uniserial,...
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2 answers
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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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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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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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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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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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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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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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2 votes
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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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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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2 answers
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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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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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2 votes
1 answer
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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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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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2 answers
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$\operatorname{Hom}_R(F,R)$ 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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$R$-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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1 vote
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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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