Apply Cayley-Hamilton's Theorem, Nakayama's Lemma and Krull's Dimension to Study the Relationship between the Pruefer's Domains and Locally Normal Rings

Abstract

This paper presents the relationship between the Pruefer's Domains and the localization of rings at maximal ideal(locally normal rings and locally valuation rings), where it is introduced some of the definitions and basic properties from commutative algebra in the first section (introduction), in the second section was presented the Cayley-Hamilton's theorem and Nakayama's lemma for finitely generated modules and the concept of the localization at prime ideal. In the third section, it is presented the relationship between Pruefer's Domain and locally valuation ring in theorem(3-2) [16,9]. In the last section it is proved in the theorem (4-3) that every locally normal ring equivalent to locally valuation ring, where it used the Krull's dimension, the concept of the primary ideal, Cayley-Hamilton's theorem and Nakayama's lemma, then it is presented a result (4-4) which shows that Pruefer's Domain equivalent to locally normal ring, by using the conditions in theorem (4-3).