Re: Prime ideals in comm. ring - HELP!

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 09/08/04 

Date: Wed, 08 Sep 2004 11:47:32 +0100

Jyrki Lahtonen wrote:

>
>> Yes, the intersection and union of a chain of ideals is an ideal.
>> Also the union of a chain of prime ideals is a prime ideal.
>> As for the intersection of a chain of prime ideals, this I see not.
>>
>> If ab in /\_i I_i, for a chain of prime ideals { I_i | i in index set }
>> then sometimes a in I_i and other times b in I_i. So how is it
>> that either a is in all I_i or b is in all I_i?
>>
> Because it is a CHAIN of ideals (think of a descending nested chain).

How would that differ from an ascending nested chain :-)

> If from some point i_1 on b is not in I_j (for j>i_1), and from some other
> point i_2 on a is not in I_j (for j>i_2), then what happens starting
> at max{i_1,i_2}? 

-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_

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