1. PRELIMINARIES

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 [39] 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,... ,

o 1

I

K dim (M. -,/M.) i a . It is possible that there is no ordinal a such

l-l

l

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-

tions.

The notion of Krull dimension has already been extended to arbi-

trary ordinals by Krause [27]. We note that for a commutative noether-

ian ring, this notion coincides with the usual "classical" Krull dimen-

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