Different definitions of the dimension of an algebra
In non-commutative algebra Krull dimension has been generalised by Gabriel & Rentschler. A decent account of it can be found in Chapter 6 of McConnell and Robson's book on non-commutative Noetherian rings.
The basic idea is as follows: An artinian module has Krull dimension 0.
A module that does not have Krull dimension 0 has Krull dimension 1 if in every infinite descending chain of submodules all but finitely many composition factors have Krull dimension 0.
A module that does not have Krull dimension 0 or 1 has Krull dimension 2 if in every infinite descending chain of submodules all but finitely many composition factors have Krull dimension 0 or 1.
The definition continues for all finite ordinals (and can be extended to all ordinals). Then the Krull dimension of a ring R is the Krull dimension of R as a module over itself.
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.
Often the most useful dimension in non-commutative algebra is the length of the minimal injective resolution of the ring as a module over itself. In many important cases this is the same as the global dimension when the latter is finite, but it is more robust in that it is finite more often. In a commutative Noetherian ring it is the same as the Krull dimension when it is finite.