Re: Ideal Theory 101 [was: is 0 prime?]

[wlawvere@buffalo.edu]

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>:

> --- 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
> 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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>=20