Re: The division lattice as a category: is 0 prime?

[Jeff Egger <jeffegger@yahoo.ca>]

--- Vaughan Pratt <pratt@cs.stanford.edu> wrote:

> I arrived at all this after Steve Vickers mentioned on the univalg
> mailing list that ring theorists define 0 to be a prime number because
> then they could define n to be prime just when the ring Z/nZ extends to
> a field.  

Um, well, for arbitrary ideals I in a commutative ring R, R/I "extends to 
a field" (or, in more common parlance, "is an integral domain") if and only 
if I is a prime ideal; hence the previous assertion can be simplified to 

  ring theorists define 0 to be a prime number because
  then they could define n to be prime just when nZ is a prime ideal. 

which doesn't seem so unreasonable.  

> This got me to wondering how this could be reflected in the
> division lattice, which has 0 at the top without however being
> considered a prime.  I personally am too old to believe that 0 is a
> prime, but I can see where a younger generation could be hoodwinked.

And I thought that every generation since Dedekind, Krull and Noether
knew that divisibility lattices are (in the general case) a red herring 
and that it is the lattice of ideals of a ring (or its opposite, if you 
prefer) which is really important.  Surely, it makes sense to fix 
terminology according to what does work in the general case.  

> Even with the above understanding however I don't see how 0 can be
> understood as just another ordinary prime, any more than bottom is just
> another ordinary number in N_*.

Although 0 can be a prime (depending on the ring under consideration), 
it is plainly never "just another ordinary prime": there is a well-known
topology on the set of prime ideals of a commutative ring which clearly
distinguishes 0 from its fellows.  Perhaps the answer to your original 
question is to take (finite-valued) sheaves on this space of primes, 
although I don't really understand your motivation.  

Cheers,
Jeff Egger.




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