ICPAM-2020, Van, Turkey, 3 - 05 September 2020, pp.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.)