Commutative Ring Closure Of The Whole Ideal

The integral closure of ideals in commutative ring originated in the1860s, has been developed into one of the most active research topics in the research of commutative algebra now。In this paper, we study the integral closure of ideals in commutative rings. First, we present a large number of examples, reflect what is the integral closure of ideals from theory and geometry intuitively。On this basis, we surveyed the intersection, product, and sum of ideals。 Secondly, in view of the regular ring, we set up a characterization of an ideal which is also integrally closed。We proved that for a regular ring R, and ideal I in ring R, I=√I, as the application of this result。We get further, if ring R is a regular ring arithmetic, every ideal is integrally closed, and under the condition of R is a semilocal ring, its inverse is right too。 Finally, we simply examine the integral closure of rings。...