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Tim Peters <tim.one@xxxxxxxxxxx> wrote in [0]:
>Bill Dubuque <wgd@xxxxxxxxxxxxxxxxxxxx> wrote in [1]:
>>
>> The set M of m>0 such that p|mb is closed under subtraction>0,
>> so all elements m of M are multiples of the smallest element d
>> (otherwise 0 < m mod d < d would be a smaller element of M).
>>
>> Next suppose that p|ab but not p|a. Hence a in M; p in M.
>> The above implies p|db where d|a and d|p, so d=1 or d=p.
>> But d=p => p|a, contra hypoth. So d=1 and p|db=b. QED
>
> The "general pattern" is useful in many contexts: suppose S is
> a set of integers and S is closed under (ordinary) subtraction.
> Then S is also closed under negation and addition; S contains 0;
> and S = {0}, or S = the set of all integer multiples of d where
> d is the smallest strictly positive integer in S.
The "general pattern" is Euclidean => PID => atoms are prime.
The above is merely a specialization of this implication to Z
with the first implication unrolled inline in descent form.
The key is to recognize the hidden ideal-theoretic structure,
as I have stressed here on many prior occasions, e.g. [2].
--Bill Dubuque
[0]
http://google.com/groups?threadm=6cCdnaDvdLNdwa3YnZ2dnUVZ_t6dnZ2d%40comcast.com
[1] google.com/groups?threadm=y8zvf17cjph.fsf%40nestle.csail.mit.edu">http://google.com/groups?threadm=y8zvf17cjph.fsf%40nestle.csail.mit.edu
[2] google.com/groups?q=author%3Adubuque+hidden+ideals">http://google.com/groups?q=author%3Adubuque+hidden+ideals
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