15 - Krull Dimension

Summary

Krull dimension is a measurement of size of a ring that has an intrinsic importance of its own and is also a useful technical tool in the theory of noetherian rings. We have already discussed in the previous chapter the “classical” Krull dimension, which originated in commutative ring theory and is defined using the prime ideals of a ring. In the noncommutative theory, we need a notion of Krull dimension that does not depend on prime ideals, but which shares many of the important properties of the classical Krull dimension for commutative rings. For instance, we would like a notion of Krull dimension that gives some useful information even for simple rings. This is done by defining a dimension on modules rather than just on rings, which has the advantage that it in some sense replaces considerations involving two-sided ideals with considerations involving only one-sided ideals. The definition now used is due to Rentschler and Gabriel and will be defined in detail in the next section. While at first sight it appears completely unrelated to the classical definition, we shall see that it coincides with the classical Krull dimension on commutative noetherian rings and, in fact, on FBN rings.