On t-reduction and t-integral closure of ideals in integral domains

Let R be an integral domain and I a nonzero ideal of R. An ideal J ⊆ I is a t-reduction of I if (J I ⁿ ) _t = (I ⁿ⁺¹ ) _t for some n ≥ 0. An element x of R is t-integral over I if there is an equation x ⁿ + a ₁ x ⁿ⁻¹ + · · · + a _n−1 x + a _n = 0 with a _i ∈ (I ⁱ ) _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 surveys recent literature which studies t-reductions and t-integral closure of ideals in arbitrary domains as well as in special contexts such as Prüfer v-multiplication domains, Noetherian domains, and pullback constructions.