--- 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. Ask a question on any topic and get answers from real people. Go to Yahoo! Answers and share what you know at http://ca.answers.yahoo.com