t-Reductions and t-integral closure of ideals
Abstract
Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JIn)t=(In+1)t for some positive integer n. An element x in R is t-integral over I if there is an equation xn + a1xn-1 +...+ an-1x + an = 0 with ai in (Ii)t for I = 1,...,n. The set of all elements that are t-integral over I is called the t-integral closure of I. This paper investigates the t-reductions and t-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well-known results, in the literature, on the integral closure of ideals and its correlation with reductions. Namely, Section 2 identifies basic properties of t-reductions of ideals and features explicit examples discriminating between the notions of reduction and t-reduction. Section 3 investigates the concept of t-integral closure of ideals, including its correlation with t-reductions. Section 4 studies the persistence and contraction of t-integral closure of ideals under ring homomorphisms. All along the paper, the main results are illustrated with original examples.