The notion of the Krull dimension of a commutative noetherian
ring, measured on chains of primes, has been used for a long time. In
[39], Rentschler and Gabriel described a Krull dimension, defined on
certain modules over noncommutative rings, and coinciding with the fa-
miliar one for finitely generated modules over a commutative noetherian
ring. (To be precise, it coincides when finite, for, unlike the origi-
nal, this Krull dimension is an ordinal.) The definition is given in
Section 1; but, briefly, for amodule to have Krull dimension a
means that any descending chain of submodules with "large" factors must
terminate, where "large" means the factors are not of Krull dimension
a . Thus an artinian module has Krull dimension 0 ; and the ring of
integers, having no proper nonartinian factors, has Krull dimension 1 .
Although not all modules have a Krull dimension, noetherian mod-
ules always do—see Section 1; and there has been much interest, in the
past few years, in the Krull dimension of noncommutative noetherian
rings and their modules, as is shown by our reference list. In this
memoir, as well as providing an exposition of much of what is known
about Krull dimension (but not all; see, for example, [18] and [39]), we
describe a perhaps surprising phenomenon. We study those rings and
modules, not necessarily noetherian, which have a Krull dimension. The
results we obtain show that merely having a Krull dimension is a strin-
gent condition with many noetherian-like consequences. (However, as
demonstrated by the examples in Section 10, rings with Krull dimension
can differ markedly from noetherian rings, even when they are commuta-
tive domains.) We expect these results, and the methods involved, to
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