Collections of an ideal: any n-absorbing ideal is strongly n-absorbing

Abstract

We show that any n-absorbing ideal must be strongly n-absorbing, which is the first of Anderson and Badawi’s three interconnected conjectures on absorbing ideals. We prove this by introducing and studying objects called maximal and semimaximal collections, which are tools for analyzing the multiplicative ideal structure of commutative rings. Furthermore, we discuss and make heavy use of previous work on absorbing ideals done by Anderson and Badawi, Choi and Walker, Laradji, and Donadze, among others. At the end of this paper, we pose three conjectures, each of which implies the last unsolved conjecture made by Anderson and Badawi.

Keywords:

• Anderson–Badawi conjectures
• collections of an ideal
• maximal collection
• n-absorbing ideal
• semimaximal collection
• strongly n-absorbing ideal

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13A15
• 13A05
• Secondary: 13F15
• 13A02

Acknowledgments

I am thankful toward Adam Logan for his remarks on a rough draft of this manuscript, and for his attentive support and advice throughout these years. I would also like to thank Ayman Badawi for directing me toward useful material, and for his remarks on my earlier (unrelated) approaches to the subject.