Use Cayley-Hamilton's Theorem and Nakayama's Lemma to Study the Relationship between the Normal Rings and Valuation Rings

Abstract

This paper presents an application of the Cayley-Hamilton's theorem and Nakayama's lemma to study  the relationship between the normal rings and 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 a general ideal about the localization of a ring at a prime ideal. In the third section, it is proven theorem (3-5) that  the localization of an unique factorization domain  at an ideal generated by a prime element from this domain will be a valuation ring. In the last section it is presented in the theorem (4-1) that every valuation ring will be a normal ring and an example shows that the inverse is not true, then use the concept of the primary ideal, local ring, Cayley-Hamilton's theorem and Nakayama's lemma in theorem (4-3) to prove that every normal ring will be a valuation ring, then it is presented a result (4-4) which shows that every non-zero ideal in normal ring is written as a power ofa maximal ideal, by using the conditions in theorem (4-3).