A Study of Relationship between the Valuation Ring, the Localization of the Ring at a Prime Ideal, and the Principle Ideals Domain

Abstract

This paper presents a study of the relationship between the Valuation Ring and both the localization of the ring at prime ideal and the Principle Ideals Domain, where it is introduced some of the definitions and properties in the first section (introduction), in the second section was presented  the definitions of evaluation ring and discrete valuation ring, then the result (2-2) shows that the valuation ring for some valuation map is a subring of the fractal (Quotient) field  and valuation ring establish itself, in addition to the theorem (2-3) provided the necessary condition and sufficient in order to be a ring valuation ring.

The relationship between the valuation ring and  localization were displayed in the third section where it was proof that the local ring of the Principle Ideals Domain at a prime ideal will be evaluation ring and each evaluation ring is containing only one maximal ideal in theorems  (3-2) and (3-3) respectively. In addition to the result (3-4) shows that each evaluation subring of integral domain has dimension "Krull's Dimension"  greaterthen this contains. The fourth section studies the relationship between the discrete valuation ring and principle ideals domain, where the theorem (4-3) proof that each discrete evaluation ring is a principle ideal domain , and vice versa it is adding some conditions and through proven (4-5).