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If $B$ is an $A$-algebra of Noetherian rings then $textAss_AM=f^-1(p):p in textAss_BM$ for a f.g. $B$-module $M.$

Noetherian module over noetherian ringNoetherian module implies Noetherian ring?$B$ is a finitely generated module over $A$. Then if $A$ is a Noetherian ring (or Artinian ring), then $B$ is a Noetherian (Artinian) ring.Prove a module is NoetherianCharacterization of right noetherian rings$A oplus M$ is Noetherian iff $A$ is Noetherian and $M$ is a f.g. $A$-moduleIf $M$ is finitely generated left module over a left noetherian ring $R$, then $M$ is a noetherian module.Localization of module of noetherian ring is f.g. but module isn't f.g.Commutative rings over which every module $M$ satisfies $mathrmAss(R/mathrmAnn; M) subseteq mathrmAss(M)$If $ operatornameAss(M)= operatornameAssh(M)$, then $M$ is a Cohen-Macaulay $R$-module?

1

$begingroup$

Let $A,B$ Noetherian rings and $f: A to B$ be a ring homomorphism. Let $M$ be a finitely generated $B$ module. Then want to show that $textAss_AM=f^-1(p):p in textAss_BM.$

So far I could prove that RHS is contained in the LHS. If $p in textAss_BM$ then $p=(0:_Bx)$ for some $x neq 0$ in $M.$ Then clearly $f^-1(p)=(0:_Ax).$ Now for the converse let $q=(0:_Az) in textAss_AM$. Then clearly $f^-1(0:_Bz)=q,$ but failed to show that $(0:_Bz) in textSpec(B).$ I need some help. Thanks.

abstract-algebra ring-theory commutative-algebra modules primary-decomposition

share|cite|improve this question

edited Mar 13 at 18:57

Bernard

123k741116

asked Mar 13 at 17:36

user371231user371231

405511

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  See stacks.math.columbia.edu/tag/05DZ.
  $endgroup$
  – Eric Wofsey
  Mar 14 at 4:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It really helps. Many thanks.
  $endgroup$
  – user371231
  Mar 14 at 17:37
  

  
  
  
  

add a comment | 

1

$begingroup$

Let $A,B$ Noetherian rings and $f: A to B$ be a ring homomorphism. Let $M$ be a finitely generated $B$ module. Then want to show that $textAss_AM=f^-1(p):p in textAss_BM.$

So far I could prove that RHS is contained in the LHS. If $p in textAss_BM$ then $p=(0:_Bx)$ for some $x neq 0$ in $M.$ Then clearly $f^-1(p)=(0:_Ax).$ Now for the converse let $q=(0:_Az) in textAss_AM$. Then clearly $f^-1(0:_Bz)=q,$ but failed to show that $(0:_Bz) in textSpec(B).$ I need some help. Thanks.

abstract-algebra ring-theory commutative-algebra modules primary-decomposition

share|cite|improve this question

edited Mar 13 at 18:57

Bernard

123k741116

asked Mar 13 at 17:36

user371231user371231

405511

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  See stacks.math.columbia.edu/tag/05DZ.
  $endgroup$
  – Eric Wofsey
  Mar 14 at 4:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It really helps. Many thanks.
  $endgroup$
  – user371231
  Mar 14 at 17:37
  

  
  
  
  

add a comment | 

$begingroup$

Let $A,B$ Noetherian rings and $f: A to B$ be a ring homomorphism. Let $M$ be a finitely generated $B$ module. Then want to show that $textAss_AM=f^-1(p):p in textAss_BM.$

So far I could prove that RHS is contained in the LHS. If $p in textAss_BM$ then $p=(0:_Bx)$ for some $x neq 0$ in $M.$ Then clearly $f^-1(p)=(0:_Ax).$ Now for the converse let $q=(0:_Az) in textAss_AM$. Then clearly $f^-1(0:_Bz)=q,$ but failed to show that $(0:_Bz) in textSpec(B).$ I need some help. Thanks.

abstract-algebra ring-theory commutative-algebra modules primary-decomposition

share|cite|improve this question

edited Mar 13 at 18:57

Bernard

123k741116

asked Mar 13 at 17:36

user371231user371231

405511

$endgroup$

share|cite|improve this question

edited Mar 13 at 18:57

Bernard

123k741116

asked Mar 13 at 17:36

user371231user371231

405511

share|cite|improve this question

edited Mar 13 at 18:57

Bernard

123k741116

asked Mar 13 at 17:36

user371231user371231

405511

edited Mar 13 at 18:57

Bernard

123k741116

edited Mar 13 at 18:57

Bernard

123k741116

asked Mar 13 at 17:36

user371231user371231

405511

asked Mar 13 at 17:36

user371231user371231

405511