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$M$ be a complete $R$-module and $N$ be a closed submodule of $M.$ Then $M/N$ is complete.

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$begingroup$

Let $M$ be a complete $R$-module with respect to the filteration ${M_n}.$ Also let $N$ be a closed submodule of $M.$ Then how can I show that $M/N$ is complete with respect to the induced filteration, i.e., ${(M/N)_n=(N+M_n)/N}.$

We should show that $M/N$ is Hausdorff and every Cauchy sequence in $M/N$ converges. Since $N= cap_{n=1}^{infty}(N+M_n)$ clearly $M/N$ is Hausdorff. But How can I show that every Cauchy sequence in $M/N$ is convergent ? I need some help. Thanks.

abstract-algebra commutative-algebra modules topological-groups

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asked 2 hours ago

user371231user371231

386511

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0

$begingroup$

Let $M$ be a complete $R$-module with respect to the filteration ${M_n}.$ Also let $N$ be a closed submodule of $M.$ Then how can I show that $M/N$ is complete with respect to the induced filteration, i.e., ${(M/N)_n=(N+M_n)/N}.$

We should show that $M/N$ is Hausdorff and every Cauchy sequence in $M/N$ converges. Since $N= cap_{n=1}^{infty}(N+M_n)$ clearly $M/N$ is Hausdorff. But How can I show that every Cauchy sequence in $M/N$ is convergent ? I need some help. Thanks.

abstract-algebra commutative-algebra modules topological-groups

share|cite|improve this question

asked 2 hours ago

user371231user371231

386511

$endgroup$

add a comment | 

$begingroup$

Let $M$ be a complete $R$-module with respect to the filteration ${M_n}.$ Also let $N$ be a closed submodule of $M.$ Then how can I show that $M/N$ is complete with respect to the induced filteration, i.e., ${(M/N)_n=(N+M_n)/N}.$

We should show that $M/N$ is Hausdorff and every Cauchy sequence in $M/N$ converges. Since $N= cap_{n=1}^{infty}(N+M_n)$ clearly $M/N$ is Hausdorff. But How can I show that every Cauchy sequence in $M/N$ is convergent ? I need some help. Thanks.

abstract-algebra commutative-algebra modules topological-groups

share|cite|improve this question

asked 2 hours ago

user371231user371231

386511

$endgroup$

share|cite|improve this question

asked 2 hours ago

user371231user371231

386511

share|cite|improve this question

asked 2 hours ago

user371231user371231

386511

asked 2 hours ago

user371231user371231

386511

asked 2 hours ago

user371231user371231

386511