$t$-reductions and $t$-integral closure of ideals

Abstract

Let $R$ be an integral domain and $I$ a non\-zero ideal of $R$. An ideal $J\subseteq I$ is a $t$-reduction of~$I$ if $(JI^{n})_{t}=(I^{n+1})_{t}$ for some integer $n\geq 0$. An ele\-ment $x\in R$ is $t$-integral over $I$ if there is an equation $x^{n}+a_{1}x^{n-1}+ \cdots +a_{n-1}x+a_{n}=0$ with $a_{i}\in (I^{i})_{t}$ for $i=1,\ldots ,n$. 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 corr\-el\-ation 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 id\-eals under ring homomorphisms. Throughout the paper, the main results are illustrated with original examples.