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On finiteness of $cup_nge 1 operatornameAss_R (R/I^n)$

$M$ is $bigcap operatornameAss(M)$-primaryUnion of Associated Primes being finite.Associated primes of a quotient module.Projective dimension zero or infinityDepth zero module and $R$-regular elementIf $P in operatornameAssM$, then $R/P subset M$.Equivalence of two definitions of primary idealsConverse of a commutative algebra theorem related to associated prime idealsPathological examples of finitely generated modulesIf $ operatornameAss(M)= operatornameAssh(M)$, then $M$ is a Cohen-Macaulay $R$-module?

1

$begingroup$

Let $I$ be an ideal of a commutative noetherian ring $R$.

How to prove that $bigcup_nge 1 operatornameAss_R (R/I^n)$ is finite ?

I am aware of Brodmann's result about Asymptotic stability of $operatornameAss_R(M/I^nM)$ for finitely generated module $M$ over commutative Noetherian ring $R$ , but I am trying to prove the claim in a direct, elementary way. I am, of course, allowed to use primary decomposition.

I can see that $Ass_R (bigoplus_n ge 1 R/I^n)=bigcup_nge 1 operatornameAss_R (R/I^n)$, however since $bigoplus_n ge 1 R/I^n$ is not finitely generated, I'm not sure if it is helpful here or not ...

Please help

ring-theory commutative-algebra modules noetherian primary-decomposition

share|cite|improve this question

edited Mar 18 at 5:59

user521337

asked Mar 16 at 9:48

user521337user521337

1,2061417

$endgroup$

add a comment | 

1

$begingroup$

Let $I$ be an ideal of a commutative noetherian ring $R$.

How to prove that $bigcup_nge 1 operatornameAss_R (R/I^n)$ is finite ?

I am aware of Brodmann's result about Asymptotic stability of $operatornameAss_R(M/I^nM)$ for finitely generated module $M$ over commutative Noetherian ring $R$ , but I am trying to prove the claim in a direct, elementary way. I am, of course, allowed to use primary decomposition.

I can see that $Ass_R (bigoplus_n ge 1 R/I^n)=bigcup_nge 1 operatornameAss_R (R/I^n)$, however since $bigoplus_n ge 1 R/I^n$ is not finitely generated, I'm not sure if it is helpful here or not ...

Please help

ring-theory commutative-algebra modules noetherian primary-decomposition

share|cite|improve this question

edited Mar 18 at 5:59

user521337

asked Mar 16 at 9:48

user521337user521337

1,2061417

$endgroup$

add a comment | 

$begingroup$

Let $I$ be an ideal of a commutative noetherian ring $R$.

How to prove that $bigcup_nge 1 operatornameAss_R (R/I^n)$ is finite ?

I am aware of Brodmann's result about Asymptotic stability of $operatornameAss_R(M/I^nM)$ for finitely generated module $M$ over commutative Noetherian ring $R$ , but I am trying to prove the claim in a direct, elementary way. I am, of course, allowed to use primary decomposition.

I can see that $Ass_R (bigoplus_n ge 1 R/I^n)=bigcup_nge 1 operatornameAss_R (R/I^n)$, however since $bigoplus_n ge 1 R/I^n$ is not finitely generated, I'm not sure if it is helpful here or not ...

Please help

ring-theory commutative-algebra modules noetherian primary-decomposition

share|cite|improve this question

edited Mar 18 at 5:59

user521337

asked Mar 16 at 9:48

user521337user521337

1,2061417

$endgroup$

share|cite|improve this question

edited Mar 18 at 5:59

user521337

asked Mar 16 at 9:48

user521337user521337

1,2061417

share|cite|improve this question

edited Mar 18 at 5:59

user521337

asked Mar 16 at 9:48

user521337user521337

1,2061417

asked Mar 16 at 9:48

user521337user521337

1,2061417

asked Mar 16 at 9:48

user521337user521337

1,2061417