An Artinian analogue of the Noetherian result on dimension of Noetherian modules

ICPAM-2020, Van, Türkiye, 3 - 05 Eylül 2020, ss.60

In this study, we present a generalization of the theorem on the Krull

dimension for Artinian modules over quasi-local rings (i.e., rings with only

one maximal ideal) to the case where the rings are not necessarily quasilocal.

Our main objective is to give an Artinian analogue of the following wellknown

Noetherian result.

Let R be a semi-local commutative Noetherian ring (where semi-local

means R has only nitely many maximal ideals). Then, for any nitely

generated R-module N, we have

dimR(N) = d(N) = (N)

where d(N) is the degree of the Hilbert polynomial associated to N, while

(N) stands for the least number of elements r1; : : : ; rn, n 2 N, of R such

that ℓR(N=(r1; : : : ; rn)N), the length of the R-module

N=(r1; : : : ; rn)N, is nite. (See for example [3, p. 98]. We also refer to

Chapter 4 of [1] for information about Hilbert polynomials.)