In this post I’d like to give a better (by which I mean category-theoretic) definition of the lattice of ideals than the standard one. We know that the lattice of ideals has meets and joins defined by intersection and sum, respectively, and that if a lattice is viewed as a category whose arrows are the order relation, then meet and join are the product and coproduct, respectively. But we also know that the lattice of radical ideals of a finitely-generated reduced integral -algebra is dual to the lattice of algebraic subsets of (and that the lattice of prime ideals is dual to the lattice of algebraic subvarieties), and there is a very general category-theoretic formalism for understanding subobjects in a category.
It turns out that this formalism reproduces the lattice of ideals of an arbitrary commutative ring – as long as we run it in the opposite category .
Edit, 2/9/10: The above claim is wrong. But let me tell you the construction I had in mind and you can judge whether it is more natural than the usual definition.
Set vs. CRing
A basic observation, which I learned from Reid Barton, is that if you want to only work in categories that behave like , then there are categorical reasons you shouldn’t work in – you should work in . For example, it turns out that in the coproduct distributes over the product instead of the other way around! The product in of two rings is just the direct product . The coproduct, on the other hand, is the tensor product (assumed to be over without qualification). And it turns out that
This is reasonable in the special case that , and it’s also not hard to prove “by hand.”
As another example, the categorical notion of “point” behaves badly in but fine in . In a category with a terminal object , a point of an object is a morphism . In the category of sets, is a one-element set, so this recovers the usual notion of point. Sometimes the point functor agrees with our intuition about what a point should be, but sometimes it doesn’t. For example, in , the category of -sets, a point is a fixed point. (Nevertheless it is an important categorical notion, for example in topoi.)
The terminal object in is the trivial ring, and there are no morphisms from the trivial ring to any nontrivial ring. This is because any morphism out of the trivial ring needs to preserve both the additive and the multiplicative identity, and in any nontrivial ring the two are distinct. However, let’s dualize: the initial object in (hence the terminal object in ) is , and morphisms into are a great notion of point. For example, if , then a morphism is precisely an integer point satisfying , in other words, a solution to a system of Diophantine equations. This is one way to motivate arithmetic geometry.
Of course, not all rings have -points (for example any ring containing a field). A generic way to fix this in arbitrary categories is to simply look at the entire functor (the “functor of points” perspective), which is justified by the Yoneda lemma. Thus, for example, a morphism is a tuple satisfying ; we call such a tuple an -point. The functor therefore knows about the solutions of this system over every commutative ring. This is an important way to think about, for example, algebraic groups, which make sense over any commutative ring.
A related way to think about this situation is in terms of coslice categories. In any category , given an object , the coslice category has objects consisting of morphisms and morphisms consisting of morphisms making the obvious commutative diagram commute. And in , the coslice category for a field is precisely the category of -algebras. In this category, the initial object is automatically and a morphism into the initial object is an -point, which we know the meaning of very well when the -algebra is finitely-generated. More generally, the coslice category is the category of -algebras, is still the initial object, and a morphism into the initial object is an -point.
A sensible notion of “subobject of ” in a category is a monomorphism . This is exactly the usual notion of subobject in , as well as the usual notion of subobject in many algebraic categories such as ; note that it does not refer to elements, only morphisms. However, we usually want to think of morphisms coming from isomorphic objects as “the same.” A standard way to do this is to construct something like the slice category over ; its objects are monics and its morphisms are morphisms making the obvious diagram commute. The universal property of monics then implies that any such morphism must itself be monic. The skeleton of this category is what we will call the category of subobjects of , and by the argument above is actually a partial order, which again agrees with our intuition from and algebraic categories; this construction is sometimes referred to as “equivalence classes of monics.”
In particular, the subobjects in are just subrings in the usual sense.
However, taking opposite categories, the subobjects in are precisely the quotient objects in , that is, equivalence classes of epimorphisms out of a ring. And these are precisely the quotients for an ideal! Edit, 2/9/10: It turns out that I was mistaken about this claim. We need to replace “epimorphism” with “extremal epimorphism”, it seems, for this to be true. From here the picture is nice. For example, the coproduct of two extremal epis is exactly , and their product is . Thus we can recover the entire structure of the lattice of ideals from the subobject construction (Edit, 2/9/10: well, part of it, anyway) applied to .
This is important because it is a totally abstract justification of using ideals as a way to think about rings geometrically. However, it doesn’t necessarily tell us that restricting our attention to prime ideals is a natural thing to do. This only happens if we restrict attention to certain subcategories. In the subcategory of reduced rings, for example, we get the lattice of radical ideals, which may be reasonably said to behave something like the lattice of subsets of the prime ideals (in other words, intuition from applies). In the smaller subcategory of finitely-generated reduced (but not necessarily integral) -algebras, we get the lattice of algebraic subsets of , and here the intuition from works out the nicest. (This is one reason not to require that “irreducible” be part of the definition of a variety.)
But the relationship between prime ideals and general ideals is more complicated in general rings. What we’re doing when we use prime ideals to talk about more general rings is hoping that the restriction to prime ideals that works in the nicest case continues to work, and while it does in fact work (in the sense that it is possible to write down a category opposite to using functions on prime ideals) and gives us some nice geometric intuition, arguably it is not the cleanest way to set up the theory, which I guess is why some people just stick to functors of points.