## Abstract

Let R be an integral domain and I a nonzero ideal of R. An ideal J ⊇ I is a t-reduction of I if (JI^{n})t = (I^{n+1})t for some integer n ≥ 0. An element x ϵ R is t-integral over I if there is an equation x^{n} + aix^{n-1} + ⋯ + a_{n-1}x + a_{n} =0 with a_{i} ϵ (I^{i})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.

Original language | English |
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Pages (from-to) | 1875-1899 |

Number of pages | 25 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 47 |

Issue number | 6 |

DOIs | |

State | Published - 2017 |

### Bibliographical note

Funding Information:This work was supported by King Fahd University of Petroleum & Minerals, DSR grant No. RG1328.

Publisher Copyright:

Copyright © 2017 Rocky Mountain Mathematics Consortium.

## Keywords

- Basic ideal
- Integral closure of an ideal
- Prüfer domain
- PvMD
- Reduction of an ideal
- T-ideal
- T-integral dependence
- T-invertibility
- T-operation
- T-reduction

## ASJC Scopus subject areas

- Mathematics (all)