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, mathrmdev;A=−infty; if A is has comparable elements but satisfies the d.c.c., then mathrmdev;A=0. In general, if alpha is an ordinal, we say that mathrmdev;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 mathcalK(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 mathcalK(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 An(k): if k has characteristic zero, then mathcalK(An(k))=n, and if k has positive characteristic, mathcalK(An(k))=2n. The book by McConnel and Robson has lots of information and references.
No comments:
Post a Comment