The Integrally Closed Properties Of Ideals And Its Application In Graph

The integral closure of ideals on commutative Noetherian ring was first defined by D. G. Northcott and D. Rees. Later, many scholars studied the integral closure of ideals and its properties on commutative Noetherian ring. They got a lot of results. In 1961, D. Rees proved that if I is an ideal of an analytically unramified local Noetherian ring then In(?)In-k for a constant K and every n≥K. In 1974, Briancon and Skoda proved, using analytic methods, that if I is an ideal in the convergent power series ring C{x1,…,xn}, then the integral closure of In is contained in I. In 1998, Aron Simis, Wolmer V. Vasconcelos and Rafael H. Villarreal studied the integral closure of monomial rings, they described the graph composed of monomial subrings, and the edges of the graph is composed of square-free monomials of degree two.Based on these research results, in this paper we investigated the completely irreducible ideal, the completely strong irreducible ideal and their integral closure. Specifically, we proved that R is a completely arithmetical ring, then every completely irreducible ideal is integral closed. Furthermore, we proved that if R is a completely arithmetical ring, every proper ideal can be written uniquely as a intersection of some mutually distinct maximal ideals of R, so every ideal of R is integral closed. As an application, we studied the integral closure of edge ideal on secure communications. A sent B the map of edge ideal which is encrypted, after B received the encrypted map, B would decrypt the original map of edge ideal according to the method for the integral closure of edge ideal. Thus they completed the safety transfer of information. It ensured the integrity and security of the information.