ac.commutative algebra - Should Krull dimension be a cardinal?

The Krull dimension, as defined by Gabriel and Rentschler, of not-necessarily commutative rings is an ordinal. See, for example, [John C. McConnell, James Christopher Robson, Lance W. Small, Noncommutative Noetherian rings].

More generally, they define the deviation of a poset $A$ as follows. If $A$ does not have comparable elements, $mathrm{dev};A=-infty$; if $A$ is has comparable elements but satisfies the d.c.c., then $mathrm{dev};A=0$. In general, if $alpha$ is an ordinal, we say that $mathrm{dev};A=alpha$ if (i) the deviation of $A$ is not an ordinal strictly less that $alpha$, and (ii) in any descending sequence of elements in $A$ all but finitely many factors (ie, the intervals of $A$ determined by the successive elements in the sequence) have deviation less that $alpha$.

Then the Gabriel-Rentschler left Krull dimension $mathcal K(R)$ of a ring $R$ is the deviation of the poset of left ideals of $R$. A poset does not necessarily have a deviation, but if $R$ is left nötherian, then $mathcal K(R)$ is defined.

A few examples: if a ring is nötherian commutative (or more generally satisfies a polynomial identity), then its G-R Krull dimension coincides with the combinatorial dimension of its prime spectrum, so in this definition extends classical one when these dimensions are finite. A non commutative example is the Weyl algebra $A_{n}(k)$: if $k$ has characteristic zero, then $mathcal K(A_n(k))=n$, and if $k$ has positive characteristic, $mathcal K(A_n(k))=2n$. The book by McConnel and Robson has lots of information and references.