Dimension Theory of Noetherian Rings

T H Lenagan

This is a series of three lectures introducing the basic ideas of dimension theory for noetherian rings and modules. The talks will be aimed at postgraduate and postdoctoral algebraists whose main interest is not in noetherian rings, but who may find that noetherian methods are sometimes needed in their own areas.

The series will begin by looking at desireable properties of dimension functions as generalizations of known ways of measuring the size of rings and modules in the artinian and commutative contexts, in order to extract the definition of a dimension function. Next, two of the most important dimensions, Krull dimension and Gelfand-Kirillov dimension, will be introduced and their basic properties developed. The series will finish with some applications of the methods introduced, chosen to illustrate typical uses of the methods.

The lectures of Vladimir Bavula, which follow this series, may well build on some of the notions developed here.

The books listed below will give an introductory idea to the type of material to be covered.

G.R. Krause and T.H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Pitman, 1985

C Nastasescu and F Van Oystaeyen, Dimensions of ring theory, Mathematics and its Applications, 36. D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1987.

J C McConnell and J C Robson, Noncommutative noetherian rings, Wiley, Chichester, 1987.