The awesome nature of Sup cannot be the reason why the Kummer functor exists, since it is merely used for=20 recording the result. The functor is "caused" rather=20 by an internal feature of the domain category C of=20 commutative rings: The category of quotient objects of any given R has a binary operation * that is neither sup nor inf even though in principle it can be=20 expressed as a combination of limits and colimits. We can call it R/ab=3DR/a *R/b but how does the=20 operation * specialize to C concretely ? Bill Quoting Jeff Egger <jeffegger@yahoo.ca>:
I don't know much about ring theory, so I could be confused about this, but I would have thought intersecting
--- Vaughan Pratt <pratt@cs.stanford.edu> wrote: the> m
could only get you the square-free ideals.=20
This is correct; there is simply no way of getting around the fact that=20 ideals form not just a lattice but carry a quantale structure derived=20 from the ring. [See my next post and the quotation below.] =20
--- 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.=20
--- 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 th> e 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=20 scattered about. Briefly, every ideal in a Noetherian ring can be=20 written as a finite intersection of _primary_ ideals, and this can=20 be made essentially unique by adding appropriate restrictions.=20
To obtain a more easily recognisable version of the Fundamental=20 Theorem of Arithmetic, it then remains to determine under what=20 circumstances a primary ideal must be a prime power. [A good=20 counter-example is Z[x,y], where the ideal (x,y^2) is primary,=20 but falls strictly between the prime ideal (x,y) and its square (x,y)^2=3D(x^2,xy,y^2).] =20
A Noetherian integral domain which does have this extra property=20 is called a Dedekind domain; examples include the ring of algebraic=20 integers w.r.t. an arbitrary number field---proving the latter result=20 (which is connected to an infamously incorrect proof of Fermat's=20 last theorem) is commonly cited as Dedekind's original motivation=20 for defining ideals.
See http://en.wikipedia.org/wiki/Primary_decomposition and http://en.wikipedia.org/wiki/Dedekind_domain for details. =20
Cheers, Jeff Egger.
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