THE CLASS OF COMMUTATIVE NOETHERIAN RINGS WITH FINITE VALUATION DIMENSION - Arifin, Samsul
A valuation ring is a commutative ring whose collection of all ideal is totally ordered by inclusion. Measures that represent how far a commutative ring deviates from being valuation is called valuation dimension. A uniserial module is a module whose collection of all submodules is totally ordered by inclusion. A measure that represents how far a module deviates from being uniserial is called the uniserial dimension. For a ring R, the uniserial dimension of an R-module R is called the valuation dimension of a ring R. Thus the notion of the valuation dimension is motivated by the notion of the universal dimension. This dissertation discusses the class of commutative Noetherian ring with finite valuation dimensions. First, we characterize the class of commutative Noetherian ring with finite valuation dimensions. In this stage, it is produced that the class of commutative Noetherian ring with finite valuation dimensions only contains the Artinian and valuation ring. Next, we study the finitely generated modules over the commutative Noetherian ring with finite valuation dimensions, specifically related to the uniserial dimension of the module. Note that in characterizing a field it can be done by reviewing the module over that field, which is a vector space. In the same way, to determine the properties of the Noetherian/Artinian ring can be done by examining the properties of the module over the Noetherian/Artinian ring. Likewise in this dissertation, we study the properties of the commutative Noetherian ring with finite valuation dimensions, but by examining the properties of the modules over the commutative Noetherian ring with finite valuation dimensions. Our main contribution in this dissertation is the characterization of commutative Noetherian ring classes with finite valuation dimensions which are done by approiii aching through the local ring. In particular, it is shown that a commutative Noetherian ring R has a finite valuation dimension if and only if R is a valuation ring or an Artinian ring. On the other hand, we give a method to determine the uniserial dimension of a finitely generated moduleM over a discrete valuation ring R by approaching the decomposition of a finitely generated module over a discrete valuation ring, both through the torsion and the torsion-free module, to explore the uniserial dimension from the module. It has been shown that the uniserial dimension of a finitely generated module M over a discrete valuation ring R is a function of the module’s elementary divisors and the rank of the non-torsion module part. Additional results in this dissertation include the valuation dimensions of a principal ideal domain, obtained through an exploration of the factor ring of the principal ideal domain. Another additional result is a Python program code for calculating the valuation and uniserial dimensions of several rings and modules.