In this section we define the Krull dimension of a ring or module,
reprove some basic results, and show how the property is affected by var-
ious algebraic operations on the ring or module. The results in the
section will be largely taken for granted thenceforth.
The Krull dimension of a module was defined by Rentschler and
Gabriel  for finite ordinals. Their definition can be extended natur-
ally to arbitrary ordinals.
Let M be a right R-module. The Krull dimension of M , which
will be denoted by K dim M , is defined by transfinite recursion as
follows: if M=0, K dim M=~l; if a is an ordinal and K dim M i a ,
then K dim M =a provided there is no infinite descending chain
M =M ^)M1 3 . . . of submodules M. such that, for i=l,2,... ,
K dim (M. -,/M.) i a . It is possible that there is no ordinal a such
that K dim M =a . In that case we say M has no Krull dimension.
The Krull dimension of a ring R, K dim R, is defined to be the
Krull dimension of the right R-module, R .
We note, by way of example, that modules of Krull dimension 0
are precisely nonzero artinian modules; and that K dim L =1 , L the
ring of integers. Several examples are given, in the final section, to
elucidate the relationship between Krull dimension and other chain condi-
The notion of Krull dimension has already been extended to arbi-
trary ordinals by Krause . We note that for a commutative noether-
ian ring, this notion coincides with the usual "classical" Krull dimen-