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Quasi chinese remainder theorem
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cliomseerg@kriocoucke.mai
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 128

PostPosted: Mon Jul 17, 2006 1:22 pm    Post subject: Quasi chinese remainder theorem Reply with quote
Dear NG,

Is it possible to find a ring R and n maximal ideals (say with n>1)
M1,..., Mn
and n points x_i in M_i, for i = 1,...,n
such that for all x in R there is a j with 1<=j<=n with

x=/= x_j mod M_j

what if the M_i's where just prime ideals?

Sincerely,
Jose Capco
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cliomseerg@kriocoucke.mai
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 128

PostPosted: Mon Jul 17, 2006 1:41 pm    Post subject: Re: Quasi chinese remainder theorem Reply with quote
Jose Capco wrote:
Quote:
Dear NG,

Is it possible to find a ring R and n maximal ideals (say with n>1)
M1,..., Mn
and n points x_i in M_i, for i = 1,...,n
such that for all x in R there is a j with 1<=j<=n with

x=/= x_j mod M_j

what if the M_i's where just prime ideals?

Sincerely,
Jose Capco

Ok.. I think I got the answer to my question by looking at Proposition
1.10 ii) of Atiyah and Mcdonald... for maximal ideals this is true <=>
for i<>j , M_i and M_j are not coprime

Sincerely,
Jose Capco
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W. Dale Hall
science forum Guru


Joined: 29 Apr 2005
Posts: 350

PostPosted: Tue Jul 18, 2006 3:39 pm    Post subject: Re: Quasi chinese remainder theorem Reply with quote
Jose Capco wrote:
Quote:
Jose Capco wrote:
Dear NG,

Is it possible to find a ring R and n maximal ideals (say with n>1)
M1,..., Mn
and n points x_i in M_i, for i = 1,...,n
such that for all x in R there is a j with 1<=j<=n with

x=/= x_j mod M_j

what if the M_i's where just prime ideals?

Sincerely,
Jose Capco

Ok.. I think I got the answer to my question by looking at Proposition
1.10 ii) of Atiyah and Mcdonald... for maximal ideals this is true <=
for i<>j , M_i and M_j are not coprime

Sincerely,
Jose Capco


I see you've answered your own question, but it seemed
important to note for you that writing the condition

x = x_j mod M_j

is silly, if x_j is actually a member of M_j, as you
seem to have stated. This may have been a typo, but
if not I thought you ought to know.

Dale.
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