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

  • 07 Dec 2020
  • Published Resarch - Mathematics

Researchers

Shawki AL Rashed

Published in

Damascus University Journal of Basic Sciences, Volume 36, Issue 1, 2020.


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, 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" greater then 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).

Keywords: Valuation Ring, Localization, and Principle Ideals Domain.

Link to read full paper

http://damascusuniversity.edu.sy/mag/asasy/FCKBIH/file/1-2020/205-227ch.pdf