# Tagged Questions

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### Show that if $R$ is principal, $N$ is pure and $Ann(x+N)= Rd$ then there exists $y \in M$ such that $x+N=y+N$ and $Ann(y)=Rd$

Let $R$ a integral domain and $M$ a $R$-module. A submodule $N$ of $M$ is pure if for all $x \in M$ and $a \in R$ such that $ax \in N$, there exists $y \in N$ such that $ax=ay$. Show that if $R$ ...
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### $M \oplus M \simeq N \oplus N$ then $M \simeq N.$

Let $M$ and $N$ be finitely generated $R$-modules where $R$ principal domain. Show that if $M \oplus M \simeq N \oplus N$ then $M \simeq N.$
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### Show that an integral domain $R$ is principal if and only if every submodule of a cyclic $R$-module is cyclic.

Good morning, I have difficulty with this problem: Show that an integral domain $R$ is principal if and only if every submodule a cyclic $R$-module is also cyclic.
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### Torsion-free but not free

The question is asking me to give an example of a finitely generated $R$-module that is torsion-free but not free. I remember in lecture, lecturer say something about the ideal $(2,X)$ in ...
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### If for $A$ a commutative nonzero ring $A^m ≅ A^n$ as $A$-modules, then $m = n$

This is the problem I need to solve: Let $A$ be a nonzero ring. Show that if $A^m ≅ A^n$, then $m = n$. The book I got this problem from suggests using the following method to solve it: Let ...
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### Show that $Ker(f)$ is finitely generated, when $f: M \rightarrow A^n$ is a surjective A-module homomorphism and M is finitely generated

This is the problem I am attempting: Let $M$ be a finitely generated A-module and $f: M \rightarrow A^n$ a surjective homomorphism. Show that $Ker(f)$ is finitely generated. Following a hint in ...
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### Prove that $J$ is an injective module.

Suppose $J$ is an $R$-module, and for every cyclic $R$-module $C$ this short exact sequence splits $$0\rightarrow J\overset{f}{\rightarrow}B\overset{g}{\rightarrow}C\rightarrow 0.$$ Prove that $J$ ...
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### decomposition of cokernel

Lets work over $\mathbb Z$ and represent maps $f:\mathbb Z^l \longrightarrow \mathbb Z^n$ as matrices. For the following matrices write an equivalent diagonal matrix. I want to write a decomposition ...
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### R-modules over a vector space

Let $V = \mathbb{C}^3$, $A$ be the matrix with column vectors $e3, e1, e2$ (where $e1, e2, e3$ are the standard basis vectors for $\mathbb{R}^3$). $R = \mathbb{F}[X]$. Let $V_a = V$ be the R module ...
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### I'm not sure with one is correct…

At the class I wrote this at my notebook: $\frac{a}{d}\equiv\frac{b}{d}\pmod{n}\Leftrightarrow a\equiv b\pmod{n}$ Assume that $d\mid{a}\wedge d\mid{b}\wedge d\mid{n}.$ this is right? or It should be ...
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### Module, vector space and abelian group

please can someone help me to prove this two properties $\bullet$ If K is a field, then the concept of K-vector space (a vector space over K) and K-module are identical. $\bullet$ The concept of a ...
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### Structure Theorem for finitely generated modules over PIDs [duplicate]

Let $A$ be a real $4 \times 4$ matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either \begin{pmatrix} ...
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### Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
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### Extension of homorphisms on a divisible R-module

Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...
Let $a$ and $b$ be two nonzero elements of a PID $R$. Prove that direct sum $R/(a)\oplus R/(b)$ is isomorphic to the direct sum $R/(u)\oplus R/(v)$, where $u=\gcd(a,b)$ and ...