t-reductions and t-integral closure of ideals

Let R be an integral domain and I a nonzero ideal of R. An ideal J ⊇ I is a t-reduction of I if (JIⁿ)t = (Iⁿ⁺¹)t for some integer n ≥ 0. An element x ϵ R is t-integral over I if there is an equation xⁿ + aix^n-1 + ⋯ + a_n-1x + a_n =0 with a_i ϵ (Iⁱ)t for i = 1, ., n. The set of all elements that are i-integral over I is called the i-integral closure of I. This paper investigates the t-reductions and i-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 i-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. Throughout the paper, the main results are illustrated with original examples.