Ideal Theory 101 [was: is 0 prime?]
--- 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
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.
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
=20
I thought that I had explained my point of view clearly enough, but apparently I haven't. If A and B are (additive) subgroups of a ring R (commutative or otherwise), then A.B is the image of the composite A @ B ---> R @ R --m-> R (where @ denotes tensor product of abelian groups, and m is the multiplication of the R, regarded as an arrow in AbGp). What is mysterious about this? We have a functor AbGp ---> Pos which maps an abelian group X to its set of subgroups; this uses only the existence of an appropriate factorisation system on AbGp. It is, in fact, also a monoidal functor with Sub(X) x Sub(Y) ---> Sub(X @ Y) defined by (A,B) |-> (the image of) A @ B ---> X @ Y. Now regarding a ring as a monoidal functor 1 ---> AbGp, we obtain a composite monoidal functor 1 ---> Pos, which is a monoidal poset. Specifically, the multiplication on Sub(R) is defined by Sub(R) x Sub(R) ---> Sub(R @ R) --Sub(m)--> Sub(R) which is exactly what I described earlier. The mystery, if there is one, is why this monoidal poset happens to be closed. My explanation is that the monoidal functor AbGp ---> Pos factors (as a monoidal functor) through the monoidal forgetful functor Sup ---> Pos. This can be easily derived from the fact that AbGp is cocomplete and @ cocontinuous in each variable; in fact, weaker hypotheses would seem to suffice. Thus, again regarding a ring as a monoidal functor 1--->AbGp, we can consider the composite monoidal functor 1--->Sup; which is nothing more nor less than a monoidal closed poset that happens to be (co)complete---and its underlying monoidal poset (i.e., the composite 1--->Sup--->Pos) is Sub(R) by definition ... mystery solved! Not quite, I hear you say: we want ideals, not arbitrary additive subgroups---but Sub(R) retains enough information of R to remember which of its elements are ideals and which are not: an ideal is an additive subgroup A such that T.A=T=A.T, where T denotes the top element of Sub(R)---namely, R itself. This is the "second stage" referred to in my "Quantale Theory 101" post, which is the process of turning a quantale into an "affine" quantale (one whose top element is also its (multiplicative) unit). In the commutative case, at least, this poses no problem whatsoever. I hope this clarifies my point of view on the Kummer functor. I don't see its existence as having anything in particular to do with commutative rings, but rather with those properties of AbGp which cause the monoidal functor AbGp--->Pos to a) exist, and b) factor through the monoidal forgetful functor Sup--->Pos ---which, as I sketched above, are not particularly rare ones. The rest is taken care of by a purely quantale-theoretic process. [But my comment about Sup being an awesome category had more to do with why the Kummer functor "should be" interesting (aside from the obvious concrete considerations), rather than why it exists.] I admit that this isn't entirely satisfying if you really are interested in ideals as representing quotients of a ring; but I do think that it is a valid perspective, nevertheless, and welcome further discussion on the topic. Cheers, Jeff. --- wlawvere@buffalo.edu wrote:
The awesome nature of Sup cannot be the reason why the Kummer functor exists, since it is merely used for recording the result. The functor is "caused" rather by an internal feature of the domain category C of 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 expressed as a combination of limits and colimits. We can call it R/ab=R/a *R/b but how does the operation * specialize to C concretely ?
Bill
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As I emphasized under (2) in my Sep29 posting, the point of view or perspective on the Kummer functor that factors it through the large category of module categories is quite interesting and useful and thoroughly understood by categorists, and so hides no "mysteries" of a general nature. Jeff has reiterated that point now in elegant detail. But my point was that another perspective, at least as important and at least as old, is perhaps not yet so well explained categorically. Categories of spaces are often analyzed in terms of algebras of functions, hence subspaces in terms of epimorphisms of algebras, (localizations for open subspaces and) regular epimorphisms for closed subspaces. Of course the corresponding congruence relations can sometimes be identified with ideals in some sense. But algebras may be something different from commutative rings, in particular there may be no (known) categories of "modules" in which they can be identified with monoids (an important example is Cinfinity spaces and algebras). Yet the concrete example of the category of commutative rings should give clues toward understanding the geometric phenomenon that another operation besides the lattice ones crops up naturally on the closed subspaces in all these categories. The understanding sought is thus primarily about these categories themselves. (The example contrasting a relief map with a flat paper one was mentioned to show that these infinitesimals are real.) "All these categories " includes many algebraic categories, but not all. For example in the simplest algebraic category (no operations), the only "resolution" of the contradiction between intersection and image is surely trivial ? On Fri Oct 5 12:10 , Jeff Egger sent:
I thought that I had explained my point of view clearly enough, but apparently I haven't.
If A and B are (additive) subgroups of a ring R (commutative or otherwise), then A.B is the image of the composite A @ B ---> R @ R --m-> R (where @ denotes tensor product of abelian groups, and m is the multiplication of the R, regarded as an arrow in AbGp). What is mysterious about this?
We have a functor AbGp ---> Pos which maps an abelian group X to its set of subgroups; this uses only the existence of an appropriate factorisation system on AbGp. It is, in fact, also a monoidal functor with Sub(X) x Sub(Y) ---> Sub(X @ Y) defined by (A,B) |-> (the image of) A @ B ---> X @ Y.
Now regarding a ring as a monoidal functor 1 ---> AbGp, we obtain a composite monoidal functor 1 ---> Pos, which is a monoidal poset. Specifically, the multiplication on Sub(R) is defined by Sub(R) x Sub(R) ---> Sub(R @ R) --Sub(m)--> Sub(R) which is exactly what I described earlier.
The mystery, if there is one, is why this monoidal poset happens to be closed. My explanation is that the monoidal functor AbGp ---> Pos factors (as a monoidal functor) through the monoidal forgetful functor Sup ---> Pos. This can be easily derived from the fact that AbGp is cocomplete and @ cocontinuous in each variable; in fact, weaker hypotheses would seem to suffice.
Thus, again regarding a ring as a monoidal functor 1--->AbGp, we can consider the composite monoidal functor 1--->Sup; which is nothing more nor less than a monoidal closed poset that happens to be (co)complete---and its underlying monoidal poset (i.e., the composite 1--->Sup--->Pos) is Sub(R) by definition ... mystery solved!
Not quite, I hear you say: we want ideals, not arbitrary additive subgroups---but Sub(R) retains enough information of R to remember which of its elements are ideals and which are not: an ideal is an additive subgroup A such that T.A=T=A.T, where T denotes the top element of Sub(R)---namely, R itself. This is the "second stage" referred to in my "Quantale Theory 101" post, which is the process of turning a quantale into an "affine" quantale (one whose top element is also its (multiplicative) unit). In the commutative case, at least, this poses no problem whatsoever.
I hope this clarifies my point of view on the Kummer functor. I don't see its existence as having anything in particular to do with commutative rings, but rather with those properties of AbGp which cause the monoidal functor AbGp--->Pos to a) exist, and b) factor through the monoidal forgetful functor Sup--->Pos ---which, as I sketched above, are not particularly rare ones. The rest is taken care of by a purely quantale-theoretic process. [But my comment about Sup being an awesome category had more to do with why the Kummer functor "should be" interesting (aside from the obvious concrete considerations), rather than why it exists.]
I admit that this isn't entirely satisfying if you really are interested in ideals as representing quotients of a ring; but I do think that it is a valid perspective, nevertheless, and welcome further discussion on the topic.
Cheers, Jeff.
--- wlawvere@buffalo.edu wrote:
The awesome nature of Sup cannot be the reason why the Kummer functor exists, since it is merely used for recording the result. The functor is "caused" rather by an internal feature of the domain category C of 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 expressed as a combination of limits and colimits. We can call it R/ab=R/a *R/b but how does the operation * specialize to C concretely ?
Bill
Get a sneak peak at messages with a handy reading pane with All new Yahoo! Mail: http://mrd.mail.yahoo.com/try_beta\?.intl=ca
wlawvere@buffalo.edu wrote:
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.
Hear, hear. As a case in point the category Div that I described, namely the division category replacing the division lattice, has the number lcm(m,n) (least common multiple) as pushout over the categorical product gcd(m,n). This pushout is not the categorical sum of m and n, which is instead the number mn. (It is hell dealing with sum and product switching around like that down below. In the upper half of Div, namely FinSet, sum is m+n and product is mn as it is in heaven.)
We can call it R/ab=3DR/a *R/b but how does the=20 operation * specialize to C concretely ?
I was wondering the same thing. I bet something good would come out of a meeting between category theorists and ring theorists on the topic of finding the right abstractions here---presumably a lot of the groundwork is already in place, much as it was for UACT in 1993, although as I recall the algebraists didn't seem in the mood at the time. Vaughan
participants (3)
-
Jeff Egger -
Vaughan Pratt -
wlawvere＠buffalo.edu