Discussions in Integral Closure of Polynomial Ideals

The main purpose of this paper is the comparison between an ideal and its integral closure and how to obtain the integral closure of ideals in different ways. We know that integral closure of a monomial ideal is again a monomial ideal and monomial ideals are associated to graphs. As graphs are widely used in computer science and cryptography in which there are only binary numbers, we introduce the associated matrix to a graph only with digits of 0 and 1. Moreover, we compare and compute the index of stability of associated prime ideals with index of stability of the integral closure of associated prime ideals.