SEMINAR: THE OSU-OU RING THEORY SEMINAR

TITLE:   Order dimension of a Noetherian ring

SPEAKER:   Hans Schoutens

                   City University of New York

DAY/DATE: FRIDAY, January 27, 2006

TIME: 4:45 P.M.

ROOM: MW 154

Abstract:  Dimension notions in algebra can be expressed as certain

lengths of chains of ideals. So is 'Krull dimension' the maximal length

of a chain of prime ideals and 'length' the maximal length of a chain

of arbitrary ideals. Traditionally, these values are only considered

when finite. However, under the Noetherianity assumption, any chain of

ideals is well-founded and hence its order-type is an ordinal. This

'ordinal length' has not been used much, and for a good reason: it

hardly distinguishes between any two rings of a fixed Krull dimension.

However, by replacing 'chain' by 'tree', we do get a much more subtle

invariant: the 'order dimension' of a Noetherian ring is the maximal

length (height) of an infinitely branching tree inside its lattice of

ideals. I will show that this defines an ordinal-valued invariant with

interesting properties; to mention just one: a one-dimensional

(commutative) domain has order dimension one if it has no singularities

and order dimension $\omega$ otherwise. I will also give some lower and

upper bounds, and some special cases in which the order dimension can

be determined exactly (although that is in general not so easy). One

can even extend this notion to non-Noetherian rings (which, for

instance, would yield that any valuation ring has order dimension one),

but I don't know yet how useful this would be.