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.
No comments:
Post a Comment