On $n$-absorbing prime ideals of commutative rings

Year 2022, , 455 - 465, 01.04.2022

Mohammed Issoual Najib Mahdou , Moutu Abdou Salam Moutui

https://doi.org/10.15672/hujms.816436

Abstract

This paper investigates the class of rings in which every $n$-absorbing ideal is a prime ideal, called $n$-AB ring, where $n$ is a positive integer. We give a characterization of an $n$-AB ring. Next, for a ring $R$, we study the concept of $\Omega \left(R\right)=\left\{{\omega }_R\right(I);I\text{is a proper ideal of}R\},$ where ${\omega }_R\left(I\right)=min\{n;I\text{is an}n\text{-absorbing}\text{ideal of}R\}$. We show that if $R$ is an Artinian ring or a Prüfer domain, then $\Omega \left(R\right)\cap N$ does not have any gaps (i.e., whenever $n\in \Omega \left(R\right)$ is a positive integer, then every positive integer below $n$ is also in $\Omega \left(R\right)$). Furthermore, we investigate rings which satisfy property (**) (i.e., rings $R$ such that for each proper ideal $I$ of $R$ with ${\omega }_R\left(I\right)<\infty$, $\omega_{R}(I)=\mid Min_R(I)\mid $, where $Mi{n}_R\left(I\right)$ denotes the set of prime ideals of $R$ minimal over $I$). We present several properties of rings that satisfy condition (**). We prove that some open conjectures which concern $n$-absorbing ideals are partially true for rings which satisfy condition (**). We apply the obtained results to trivial ring extensions.

Keywords

n-absorbing ideal, prime ideal, primary ideal, Noetherian ring, Artinian ring, Prüfer ring

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