## Rocky Mountain Journal of Mathematics

- Rocky Mountain J. Math.
- Volume 47, Number 6 (2017), 1875-1899.

### $t$-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.

#### Article information

**Source**

Rocky Mountain J. Math., Volume 47, Number 6 (2017), 1875-1899.

**Dates**

First available in Project Euclid: 21 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.rmjm/1511254956

**Digital Object Identifier**

doi:10.1216/RMJ-2017-47-6-1875

**Mathematical Reviews number (MathSciNet)**

MR3725248

**Zentralblatt MATH identifier**

06816574

**Subjects**

Primary: 13A15: Ideals; multiplicative ideal theory 13A18: Valuations and their generalizations [See also 12J20] 13C20: Class groups [See also 11R29] 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13G05: Integral domains

**Keywords**

$t$-operation $t$-ideal $t$-invertibility P$v$MD Prüfer domain reduction of an ideal integral closure of an ideal $t$-reduction $t$-integral dependence basic ideal

#### Citation

Kabbaj, S.; Kadri, A. $t$-reductions and $t$-integral closure of ideals. Rocky Mountain J. Math. 47 (2017), no. 6, 1875--1899. doi:10.1216/RMJ-2017-47-6-1875. https://projecteuclid.org/euclid.rmjm/1511254956