I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:


* The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
* The Krull dimension, based on chains of prime ideals.
* The transcendence degree of the fraction field of A over F.


According to Artin, GK dimension is the most robust notion because it applies to certain noncommutative algebras. And in the noncommutative setting one can't form fraction fields, so the transcendence degree is out of the question (right?).


But Krull dimension has the advantage that it applies to arbitrary rings. As far as I can tell, the definition of GK dimension only applies to algebras. So: as a matter of pedagogy, which notion of dimension is most appropriate for what applications? Which is easiest to prove things about when?



Different definitions of the dimension of an algebra
In a non-commutative ring, you need to be careful with what you even mean by a prime ideal, and usually there are very few two-sided ideals you might call prime. Oh, and even in the cases when there is a nice ring of fractions, it won't be a field, and so transcedence degree is still bad.


My personal favorite notion of dimension is 'global dimension', the maximum projective dimension of any module of the ring. This concept exists for any ring, and in fact for any abelian category (though, if there aren't enough projectives, you need to play with the definition). The only problem is that it can often be infinity, even for relatively mild rings, like C[x]/x^2. It still makes for a pretty good theory of 'smooth dimension', however.


From a conceptual perspective, Krull dimension seems best suited for geometric perspectives, since it is measuring chains of irreducible closed subsets. The easiest times to work with Krull dimension is when you are in a Cohen-Macaulay ring, and then Krull dimension is equivalent to depth, which is easier to prove things about, since you only need to produce a maximal regular sequence.


