Ideal Theory 101 [was: is 0 prime?]

[Jeff Egger <jeffegger@yahoo.ca>]

--- Vaughan Pratt <pratt@cs.stanford.edu> wrote:
> I don't know much about ring theory, so I
> could be confused about this, but I would have thought intersecting them
> could only get you the square-free ideals. 

This is correct; there is simply no way of getting around the fact that 
ideals form not just a lattice but carry a quantale structure derived 
from the ring.  [See my next post and the quotation below.]  

--- Bill Lawvere <wlawvere@buffalo.edu> wrote:
> The ideal product under discussion is a key
> ingredient in a construction of unions of subspaces that takes into
> account the clashes. 

--- Vaughan Pratt <pratt@cs.stanford.edu> wrote:
> Starting from the prime
> power ideals takes care of that but what's the trick for getting all the
> ideals from just the prime ideals?  The category Div was my suggestion
> for that, but if there's a more standard approach in ring theory I'd be
> happy to use that instead (or at least be aware of it---Div is starting
> to grow on me).

I'd point you to Wikipedia, only the relevant articles are somewhat 
scattered about.  Briefly, every ideal in a Noetherian ring can be 
written as a finite intersection of _primary_ ideals, and this can 
be made essentially unique by adding appropriate restrictions. 

To obtain a more easily recognisable version of the Fundamental 
Theorem of Arithmetic, it then remains to determine under what 
circumstances a primary ideal must be a prime power.  [A good 
counter-example is Z[x,y], where the ideal (x,y^2) is primary, 
but falls strictly between the prime ideal (x,y) and its square
(x,y)^2=(x^2,xy,y^2).]  

A Noetherian integral domain which does have this extra property 
is called a Dedekind domain; examples include the ring of algebraic 
integers w.r.t. an arbitrary number field---proving the latter result 
(which is connected to an infamously incorrect proof of Fermat's 
last theorem) is commonly cited as Dedekind's original motivation 
for defining ideals.

See http://en.wikipedia.org/wiki/Primary_decomposition
and http://en.wikipedia.org/wiki/Dedekind_domain
for details.  

Cheers,
Jeff Egger.




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