t-Reductions and t-integral closure of ideals

TL;DR

This paper explores the concepts of t-reduction and t-integral closure of ideals in integral domains, establishing analogues of classical results and analyzing their properties and behavior under ring homomorphisms.

Contribution

It introduces and studies t-reduction and t-integral closure, providing new insights and examples that extend classical ideal theory to the t-operation context.

Findings

Identifies basic properties of t-reductions and distinguishes them from reductions.

Establishes relationships between t-integral closure and t-reductions.

Analyzes how t-integral closure behaves under ring homomorphisms.

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 (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in 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} in (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 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.