--- 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. Get news delivered with the All new Yahoo! Mail. Enjoy RSS feeds right on your Mail page. Start today at http://mrd.mail.yahoo.com/try_beta?.intl=ca